Mostly, it's things that are left out. Ever since Prisoner of Azkaban, the films have not been showing all of Harry's classes, preferring instead to take only enough time to highlight the teacher's we haven't seen before. As the only new teacher we meet is Professor Slughorn, only his Potions class is shown.
The film leaves out the book's opening scene, with Cornelius Fudge meeting the Muggle Prime Minister (presumably Tony Blair?). Apparently, the Muggle Minister has already had several short meetings with Fudge in the past, whenever something from the wizarding world affected the Muggle world; this time, Fudge has come to announce his resignation from his post (in light of his failure to seriously assess the Voldemort threat in the previous year), and introducing the new Minister for Magic, Rufus Scrimgeour. Neither Fudge nor Scrimgeour feature in this movie; Scrimgeour (Bill Nighy) was introduced in the next film. Instead, the opening scene of the movie shows the Death Eaters raiding houses, kidnapping Ollivander and committing terrorist acts, facts that are mentioned in the book but not shown.
The scene revealing Severus Snape conspiring with Bellatrix Lestrange and Narcissa Malfoy is more or less the same in the book, but longer. Bellatrix produces many arguments for not trusting Snape, but he has many valid explanation for once betraying the Dark Lord, and turning spy for Voldemort again.
In the book, Albus Dumbledore actually visits Harry at his house at Privet Drive instead of the underground station. This makes for a very awkward encounter between Dumbledore and the Dursleys. He announces that the magic protection Harry receives from his aunt will be wearing off, since Harry will come of age this year (as disclosed in the book, but not the movie Harry Potter and the Order of the Phoenix, when Harry's mother sacrificed herself for him, her blood gave him the ultimate protection from death; as aunt Petunia is her sister, her blood provides the same protection as long as Harry is in her vicinity, which is why he was always sent back to the Dursleys during holidays). With Sirius dead and Harry named his heir, Dumbledore claims that Sirius' old house at Grimmauld Place should now legally be Harry's. He tests this by summoning Kreacher the House Elf, who indeed (begrudgingly) recognizes Harry as his master, making Grimmauld Place Harry's new residence for the holidays as well as the continuing headquarters for the Order of the Phoenix. Harry sends Kreacher off to Hogwarts, where he can make himself useful and the other Elves can keep an eye on him. Dumbledore and Harry then proceed to professor Slughorn's house after this.
Harry's stay in the Weasley's home is longer in the book, because Bill Weasley and Fleur Delacour are there as well, preparing for their wedding (which will occur in the next book and movie). Mrs Weasley, Ginny and Hermione are quite annoyed by Fleur's constant presence and continuous nagging about the marriage, going as far as referring to her as "Phlegm".
During the visit to Diagon Alley, Harry, Ron and Hermione have a very grim run-in with Draco Malfoy and his mother, which almost ends in a duel. They follow Draco to the Borgins and Burke's store in Knockturn Alley, where they notice Draco showing the shopkeeper something (hidden from their view) that greatly terrifies the shopkeeper. Hermione tries to find out what Draco was interested in, but she gets thrown out of the shop. Harry is convinced that Draco had a Dark Mark on his arm which frightened the shopkeeper, meaning he has taken his father's place as Death Eater in Lord Voldemort's service. Harry voices this suspicion to his friends and Mr. Weasley, but no one seems to be particularly convinced.
During the train ride to Hogwarts, Horace Slughorn already organises one of his Slug Club meetings in the train (in the movie, this meeting is held later on, in Hogwarts). In the book it is actually Nymphadora Tonks, being stationed at Hogwarts to protect the students, who finds Harry's paralysed body inside the train and fixes his nose, not Luna Lovegood. While escorting Harry to the school, she sends out a Patronus as a messenger. Harry is then taken into the school by professor Snape (who deducts points from Gryffindor house for Harry being late), who notes that Tonk's Patronus has changed form after her grief over Sirius Black's death (a detail explained in the next book).
The most drastic change from the book concerns an expansion of the subplot where Harry and Dumbledore have private sessions, in which they speculate about Voldemort's complete past and immortality. The book features several more "flashback scenes" with the Pensieve than shown in the movie. The first one is a memory from Bob Ogden, a Ministery of Magic official, and his run-in with an empoverished wizard called Marvolo Gaunt. Marvolo lived in a dilapidated shack with his son Morfin and daughter Merope. He was very proud of his pure-blood ancestry, which he claimed went all the way back to Salazar Slytherin; a fact he loudly boasted by showing his black family ring (later used by Voldemort as a Horcrux, and revealed to contain the Resurrection Stone). Merope is revealed to be Voldemort's future mother; she had taken a liking to a young Muggle living nearby, Tom Riddle Sr. Enraged by this, Marvolo and Morfin attacked her, which earned them both a sentence in Azkaban. Dumbledore speculates that Merope, freed from her family's tyranny, used a love potion on Riddle, married him and got pregnant.
The character of Mundungus Fletcher, a petty low-life criminal, who was introduced in the previous novel as a member of the Order, was also written out of this adaptation (he is introduced in the next movie, Harry Potter and the Deathly Hallows: Part 1 instead). Harry catches him in Hogsmeade as he is stealing heirlooms from Sirius' old house, but Fletcher quickly Disapparates.
During Harry's second session with Dumbledore, he shows him two memories. The first is from Caractacus Burke (one of the founders of the Borgins and Burkes store), showing Merope Gaunt pregnant and desperate for money, selling one of her heirlooms, the Slytherin locket (also seen later), for a bargain price. Dumbledore surmises that she made the mistake to think that Riddle Sr. would still love her without the love potion; but as soon as the enchantment was over, Riddle came to his senses, left her and his unborn son, and returned to his parents' home. The second memory is Dumbledore's own, from when he visited the orphanage (which is in the movie); Dumbledore's talk with the director is much longer, revealing much more about the circumstances of Voldemort's birth. Harry learns that Merope gave birth to Tom Riddle Jr. and died within the hour, having lost the will to live after Riddle Sr. left her. The director also reveals that during a field trip to the coast, young Tom once lured two students into a cave and scared them (presumably with magic). This cave is later revealed to contain the locket Horcrux.
In the movie, Ron gets to play Quidditch for the first time, whereas in the book, he has already been playing it since last year and Harry has become Quidditch captain (the subplot of Ron's introduction into Quidditch was carried over from the previous book). However, Ron suffers major performance issues over the course of the book, when he starts to clash with Hermione over their respective relationships, and with his sister Ginny, who is openly dating Dean Thomas. This latter part also greatly affects Harry, as he finds out that his feelings for Ginny are getting stronger.
A scene only in the movie is the Death Eater attack on the Burrow, the Weasley's home. The book features a scene where Harry is enjoying the Christmas holidays at the Burrow, when he gets a visit from Scrimgeour. Scrimgeour requests that Harry works together with the Ministry of Magic to boost morale in the wizarding world. However, Harry refuses, given the way the Ministry treated him the year before, and because they are making random arrests in their search for Death Eaters.
During the third session, Harry learns from Dumbledore that Voldemort had gathered a group of students during his time in Hogwarts, who shared his views on pure-blood wizardry and would become his Death Eaters later on. Voldemort had used the holidays from Hogwarts to trace his maternal family, and had visited his uncle Morfin, the only one alive at the time (his grandfather Mavolo having died in Azkaban). Morfin, however, had a different recollection from what really happened, so Dumbledore visited him just prior to his death, and through some complicated Legimency, he was able to piece together what had happened exactly: Morfin had told Tom Riddle Jr. that Tom Riddle Sr. was a Muggle, making Tom a half-blood wizard. Enraged by this and his father's abandonment, Voldemort had gone out to kill Riddle Sr. and his Muggle grandparents. He had then stunned Morfin and modified his memory, making Morfin believe that he himself committed the murders. Voldemort also took Morfin's black family ring, made it into a Horcrux, and hid it inside the house (Dumbledore found it between the events of Order of the Phoenix and Half-Blood Prince). Morfin went back to Azkaban for life, but died before Dumbledore could get him released. The second memory shown to Harry is the modified memory from Slughorn, where young Voldemort inquires about Horcruxes. In the movie, the course of the events was simply altered and the word "Horcrux" muted, while in the book, the memory becomes foggy and Slughorn has inserted an alternate voice-over by himself where he had made the modifications.
Dobby the House Elf has actually appeared in every book since Chamber of Secrets, with the exception of Prisoner of Azkaban. Harry's suspicion of Draco Malfoy leads him to task Dobby and Kreacher, the Elf he inherited from Sirius, with spying on Malfoy.
Harry has a fourth lesson with Dumbledore, where he is forced to admit that he still has not retrieved Slughorn's unaltered memory. Dumbledore reaffirms the importance of the complete memory, and shows Harry two more memories. The first is from the House Elf of a woman called Hepzibah Smith, who showed Voldemort the Slytherin locket, previously owned by his mother, and a cup owned by Helga Hufflepuff. According to Dumbledore, Voldemort stole both items, killed Smith and framed the murder on her House Elf. The next memory is from Dumbledore himself, showing the adult Voldemort applying for the job of Defence Against The Dark Arts teacher. However, as Dumbledore expected that Voldemort only wanted this job to recruit more followers to his dark cause at the school, he refused, after which an angry Voldemort left. Dumbledore implies that Voldemort cursed the post of Defence Against The Dark Arts teacher; no one has been able to keep the position for more than a year since then.
Harry receives a letter from Hagrid in the morning, announcing Aragog's funeral. Harry needs to be a bit more persuasive in getting Slughorn to give him the untampered memory than in the movie, as Slughorn is quite unwilling; in the movie, Slughorn spontaneously relates the story of how he received a pet fish from Lily Potter (not in the book), and guilt-ridden, he gives Harry the memory.
The fifth lesson involves Slughorn's untampered memory, as shown in the film as well. Afterwards, Dumbledore and Harry discuss the former Horcruxes; Tom Riddle's diary and the black ring of Marvolo Gaunt. The diary had been given in the care of Lucius Malfoy just before Voldemort was beaten the first time, and Malfoy, not knowing it was a Horcrux, later used it carelessly to get the Chamber of Secrets opened in the second movie and book. The black ring had another curse on it; when Dumbledore destroyed it, it poisened him, and professor Snape was just able to contain this spreading curse in Dumbledore's hand. Dumbledore mentions that four Horcruxes remain, assuming the Hufflepuff cup and the Slytherin locket are among them; the third may be an object that once belonged to Rowena Ravenclaw; Voldemort's snake Nagini is a likely option for the fourth Horcrux. None of this speculation is in the movie, so in the final two movies, Harry has no clue what the Horcruxes may be or what they look like beforehand.
Upon learning that Ginny and Dean split up, Harry is relieved; however, having seen Ron's reaction to Ginny's flirt with Dean, he is afraid to make a move, fearing it may cost him Ron's friendship.
After Harry wounds Draco in the bathroom, Snape uses Legimency on Harry, to find out that his Potions book is the source of the Sectumsempra curse. He orders Harry to bring to him all his school books. Harry quickly hides the Half-Blood Prince's book in the Room of Requirement, where he also sees a silver diadem (which plays a role in the next book), and gives Snape Ron's copy instead. Snape, however, knows Harry is trying to trick him, and gives him detention every Saturday until the end of the year, causing Harry to miss the Quidditch finale. However, Gryffindor team unexpectedly wins the match. In the heat of the celebration, Harry and Ginny kiss, and Ron seems to be fine with it.
Hagrid relates to Harry a conversation he overheard between Dumbledore and Snape, where Snape mentions that he "doesn't want to do this anymore." In the movie, it is Harry himself who witnesses this conversation on the astronomy tower before he departs with Dumbledore on the Horcrux quest.
Just before Harry meets up with Dumbledore to go in search of Horcruxes, he meets Professor Trelawney. In a moment of liquor-induced candor, she admits that while she was making the prophecy about Harry and Voldemort to Dumbledore 16 years ago, it was partially overheard by Severus Snape. Harry quickly concludes that it must have been Snape who had warned Voldemort, who in turn killed Harry's parents. Harry angrily berates Dumbledore for withholding such important information, but Dumbledore assures Harry that Snape felt immense remorse for his act; knowing that Voldemort would go after Harry's mother, he immediately abandoned the Death Eaters and has been a trusted member of the Order since. With Dumbledore unwilling to listen to any of Harry's suspicions about both Snape and Malfoy, Harry warns Ron and Hermione before leaving Hogwarts for the Horcrux search, that Malfoy may take advantage of the headmaster's absence to make a move, and gives them the rest of his Felix Felicis and the Marauder's Map.
Dumbledore takes Harry to Hogsmeade before apparating out, since Hogwarts and its grounds are magically protected from directly apparating and disapparating. In the movie, Dumbledore states that the Headmaster enjoys special privileges that allow him to apparate in and out of Hogwarts.
After retrieving the (fake) locket, Harry and Dumbledore apparate back to Hogsmeade, where they see the Dark Mark over Hogwarts. Madam Rosemertha gives them broomsticks with which they fly to Hogwarts. A weakened Dumbledore needs to make a couple of spells to cancel the magical protection that normally prevents people from flying in.
After Disarming Dumbledore, Draco admits that he had used the Imperius Curse on Madam Rosemerta so that she would deliver the cursed necklace, and poison the wine meant for Dumbledore. Snape is not present in the book prior to the departure of Harry and Dumbledore, and Harry is hidden under his invisibility cloak upon their return, so Snape does not encounter him at either time. Harry is unable to aid Dumbledore because Dumbledore himself paralyzes Harry with a Stunning spell. Dumbledore's death releases Harry's stun. In the movie, Snape sees Harry underneath the top floor and urges him to be quiet, before going upstairs to kill Dumbledore.
The climax of the film has also been altered: Bellatrix Lestrange leads the Death Eaters into Hogwarts, while in the book, she only appears during the scene in which Snape takes his Unbreakable Vow (possibly she was included as she is the only Death Eater, other than Lucius Malfoy, that audiences would be familiar with. Her presence also makes sense, given that Draco is her nephew). Several more Death Eaters enter Hogwarts in the book, but they meet with heavy resistance from teachers, members of the Order, and the students who had been member of Dumbledore's Army in Order of the Phoenix. The Death Eaters try to stun Hagrid, only to find out that magic spells hardly have any effect on him (since he is massive and half-Giant). Together the Death Eaters are driven out of the castle, whereas in the movie, they leave the castle unhindered (this entire battle was omitted as Deathly Hallows part II has a similar climax).
In the book, upon finding Dumbledore's body, Harry immediately opens up the locket, finding the message from R.A.B. directed at Voldemort (in the movie he is not shown doing this). Harry is devastated to learn that their mission to retrieve the locket was all in vain. Even worse, he feels immense guilt over forcing Dumbledore to drink the poison, because without it, Dumbledore probably would not have been easily defeated by the Death Eaters. In the movie, all students and teachers illuminate their wands to drive away the Dark Mark over the castle (not in the book).
The book describes the aftermath of the battle; one Death Eater was killed, and Bill Weasley, Ron's oldest brother, has been permanently scarred by werewolf Fenrhir Greyback, though since Greyback was in Human form when he attacked Bill, Bill will not become a full werewolf despite gaining some werewolf traits. Fleur says she will love him anyway, and refuses to delay her wedding. Ginny, Ron and Hermione were miraculously spared from harm due to drinking Harry's Felix Felicis. Dumbledore's pet Phoenix, Fawkes, is heard singing a last lament before leaving Hogwarts forever. McGonagall asks Harry what he and Dumbledore had been doing, but as promised to Dumbledore, he cannot reveal it.
Hermione discovers why Snape calls himself "The Half-Blood Prince" when she finds an article about Snape's mother, Eileen Prince, who was a witch. Since Snape's father was a Muggle, he called himself the Half-Blood Prince, as he was, in fact, a half-blood wizard.
Dumbledore is buried in a white tomb on Hogwarts' grounds in the presence of a massive crowd of wizards and witches, many of them old acquaintances from previous books (Professor Umbridge among them, much to Harry's anger). Afterwards, Harry breaks up his relationship with Ginny, citing that he is a target by the Death eaters now, and would only endanger her; she understands. Harry once again meets Scrimgeour, but again refuses to cooperate with the Ministry; he tells Ron and Hermione that he does not intend to return to Hogwarts next year, instead going on a search for the remaining Horcruxes. Ron and Hermione agree to go with him.