(at around 2 mins) During the destruction of the Millennium Bridge, to begin with the bridge is full of people, including a lot of school children, walking across. As the bridge begins to wobble, people start to trip up and fall down. When we see the bridge from above, as it is destroyed seconds later, there aren't any people on it.
(at around 2h 5 mins) After Harry scoops up a sip of potion from the basin, there is still a scoopful left in the basin. However, after Dumbledore drinks it, the basin appears to be empty and Harry grabs the locket from the empty basin.
The potions book which Harry gets from Slughorn's cupboard has the words "This book is property of the Half-Blood Prince" clearly handwritten on the first page. In a later shot the handwriting changes to a more print-like font.
(at around 36 mins) When Harry's potions class is brewing the Draught of Living Death, right as Seamus' potion blows up in his face, Dean Thomas' head instantly changes position, clearly as a result of a cut.
(at around 8 mins) When Dumbledore puts the Muggle house "back in order", the clock clearly shows the time as 10:35 p.m. However, the waitress Harry meets at the sandwich shop said she'd be off at "11pm" but Harry and Dumbledore see her and her companion closing up the shop before they leave the train station.
The spell that Snapes uses to heal Malfoy is "Vulnera Sanentur". It heals deep wounds. It however does not clean blood stains from clothing.
When Harry first has the Half Blood Prince's potions book, his signature is in the front of the book. When Ginny takes it from him, she reads his signature from the back. However, the book may have been signed at the back and the front as later in the film we see the words written in two different styles suggesting multiple instances.
In 1 or 2 scenes in the dining hall, Lavender Brown is shown sitting at a different table across from Ron Weasley. They were both in Gryffindor, so they should be at the same table. However, it is not uncommon for students to be seated at other Houses' tables. Lavender was avoiding Ron and chose not to sit with Gryffindor.
Dumbledore has Riddle's diary in his possession, even though Harry gave the book to Lucius Malfoy at the end of Harry Potter and the Chamber of Secrets (2002). However, Lucius gave it to Dobby, inadvertently freeing him as Harry had hidden his sock between the pages. Dobby then dropped the book to rescue Harry from Lucius's wrath. The book was left on the floor, meaning that someone could return the book to the headmaster's office. It is doubtful Lucius would have retrieved the book as he would have realised Harry only gave it to him to trick him and, presuming he knew it was a Horcrux, would also have realised it was worthless now that it had been destroyed.
Some have suggested that Aragog is a great deal smaller in this movie than he was in Harry Potter and the Chamber of Secrets (2002), but this is only true to an extent, as spiders generally shrivel up pretty fast after dying. Add to this the fact that the people around him this time around are quite a bit taller, plus the angles at which we saw him in his first appearance really complementing his stature while alive, and this becomes a very reasonable change which doesn't indicate any mistake or error by the filmmakers.
According to Harry Potter and the Order of the Phoenix (2007), you need to know the purpose of the Room of Requirement to be able to access it. Harry and Ginny were looking for a place to hide the book, so the Room allowed them access.
(at around 1h 45 mins) In the scene with Hagrid and Professor Slughorn, you can see them both moving their hands when talking. Slughorn's hand often disappears/seems to go right through Hagrid's arm. They were filmed separately to exaggerate Hagrid's height.
(at around 53 mins) When Professor McGonagall is examining the cursed necklace, she is standing in front of a window. During several of the shots you can see right through her.
Harry's glasses change to having lens to having no lens several times during the movie. (The filmmakers had that happen so the lenses couldn't reflect the camera while looking at Harry).
When Professor McGonagall sends Harry Hermione and Ron back to their Gryffindor dormitories, there is no scar on Harry's forehead.
(at around 5 mins) At the beginning, when Harry and Dumbledore first Disapparate, there is a full shot of them walking up to the entrance of the house. In this full shot, you can see Dumbledore's shoes sticking out of his robe as he walks, only he is actually wearing those bags on his shoes, which the cast wear in between takes so as not to damage the set floors.
Considering the war going on in the wizard world, Felix Felicis should be commonly available, but it's used only twice in the entire story.
The opening scene which is set in 1996, shows the Millennium Bridge getting destroyed. The bridge started to get built in 1998 and was completed in 2000.
(at around 1h 3 mins) During the Quidditch Match, and the celebration afterwards in the Gryffindor Common room, you can see that the students are chanting "Weasley is our king!". However, the audio track we hear is "Weasley! Weasley! Weasley!" over and over.
(at around 12 mins) When Harry and Dumbledore disapparate after leaving Slughorn, the sound of disapparation is heard before Harry actually takes Dumbledore's arm.
Just before Dumbledore and Harry go on their Horcrux mission, the silhouette of a stage hand can be seen in the clock tower ducking out of sight when he realizes he's on camera.
In a shot of the Weasley's house during what is supposed to be winter, the constellation Scorpio is clearly visible in the sky. Scorpio is never visible in Britain in winter, only the summer.
(at around 57 mins) During Harry's first visit to the Weasley's house in Harry Potter and the Chamber of Secrets (2002), a close up of the clock shows an option for the family to be at the dentist. At a Slug Club meeting, Hermione has to explain to the group what a dentist does, because that occupation doesn't exist in the wizarding world.
(at around 45 mins) Why does Cormac need Ron to introduce him to Hermione? They're all Gryffindors, and surely would've bumped into each other in the Gryffindor Common Room.
According to Dumbledore, the ring that once was a Horcrux is already destroyed by the time he presents it to Harry. If destroyed, it shouldn't act like it's alive, moving by itself for example.
Harry Potter is the Chosen One, the most wanted target of Voldemort, but he is allowed to hang out in the London Underground for an entire summer.
In a flashback, Tom Riddle asks Horace Slughorn about having multiple Horcruxes. However, according to the lore added by J.K. Rowling herself, when he started wearing it, it was already turned into his SECOND Horcrux (the first being the diary destroyed in the second film). He is already on the way to his goal of six, and he has no way back. This raises the question why he would need Slughorn's opinion now.
(at around 36 mins) During Professor Slughorn's lesson, the students are told to make a Draught of Living Death, the recipe for which "can be found on page 10". During the lesson, it is quite clear that Hermione has her book open somewhere in the middle (circa page 200).
(at around 48 mins) When Harry opens the Prince's book after the scene which Ginny takes the book from him and reads aloud "This book is property of the Half-Blood Prince", the camera goes through pages of the book and there's a page in which we see "Sectumpsempra. For Enemies". The first word is "Sectumsempra" not "SectumPsempra".
After Ron is released from the infirmary after drinking the poisoned mead, he, Harry, and Hermione are in the Great Hall at a meal. Ron notes that his now-ex-girlfriend Lavender is glaring at him angrily from a table two rows away. However, when students gather for meals, they sit at their assigned house table. Gryffendors sit at the long table second from the left, where Ron, Harry, and Hermione are seated. Lavender (and other Gryffendors) are seated at the right-most table, which has previously been seen to be the Slytherin house table.
Although students mostly ignore the Hogwarts dress code out of class, it's still out of character for Draco Malfoy, a pure-blood supremacist and Muggle-hater, to wear a Muggle suit instead of wizard robes.
(at around 1h 55 mins) After Dumbledore shows Harry Tom Riddle's diary, he tells Harry the diary is a Horcrux, but he should have said 'was' as the diary was destroyed in Harry Potter and the Chamber of Secrets (2002), meaning it is no longer a Horcrux.