The movie covers most of the book. Some scenes from the book are compressed or placed somewhere else in the story, but there are some notable omissions, listed below as they appear in the book:
The book has a different opening, which introduces Vernon and Petunia Dursley before Harry is put on their doorstep. Vernon has some strange encounters on the streets with weirdly dressed people who are in the midst of celebrating the downfall of "You-know-who". He also finds out that he has a nephew called Harry, the son of his wife's estranged sister.
Harry's early years are mentioned in more detail, especially the endless taunts from Dudley, and the dull times he had when the Dursleys dumped him with Mrs. Figg, the next-door neighbour. Harry sometimes even seems to remember vaguely how he got his scar. After Harry receives letters from Hogwarts directed at his cupboard, his uncle and aunt move him to 'Dudley's second bedroom' in a futile effort to stop the letters. The entire family even moves to a hotel to avoid the letters, but they find their way even there.
Hagrid mentions he isn't allowed to do magic since he got expelled from Hogwarts in his third year, but he refuses to elaborate. In Diagon Alley, Harry has an early meeting with Draco Malfoy while he is buying robes, but Draco does not properly introduce himself until the train voyage to Hogwarts. Harry needs to wait out the beginning of the school year at the Dursleys; Uncle Vernon takes him to the train station in London, but only because Dudley needs surgery there in order to have his pig's tail removed. Ginny Weasley is already introduced at the station in the movie, but not in the book. During the train ride, Harry immediately notices Malfoy's disdain for Ron and his family (Malfoy even tries to steal candy from Ron, but Ron's rat Scabbers intervenes), and declines his friendship. In the film Harry instead first met Malfoy at Hogwarts before the Sorting.
Apart from the other house spirits, there is another particularly nasty poltergeist called Peeves, who gets his pleasure from pulling all sorts of pranks with the students and Mr Filch. Peeves' only fear is for the Bloody Baron, though he makes a point to be on somewhat good behavior around Professor Dumbledore (Harry later uses this information to his advantage when he is under the Invisibility Cloak and pretends to be the Bloody Baron who turned himself invisible). The book also contains an additional teacher, Professor Binns, who is a ghost (he once fell asleep next to a burning fire) and gives excessively dull lessons in History of Magic. Snape's disdain for Harry is illustrated in more detail, as he manages to blame Harry for everything, punish him for every petty little thing, and deducts points from Gryffindor at almost any opportunity.
Neville is also a more prominent figure in the books as he usually accompanies Ron and Harry; in the movie, Hermione is actually used in place of Neville in some cases. After their first flying lesson, Draco Malfoy challenges Harry to a wizarding duel at midnight, which Harry and Ron accept. They sneak out that night but find that Malfoy tricked them in an attempt to get them in trouble. On their way back, they run into Neville. The three get redirected to Gryffindor Tower and end up going into the third floor corridor which is guarded by Fluffy (in the movie, it is a suddenly changing staircase that leads them there). Dismayed that his plan to rat out Harry has failed, Draco tries again by alerting the teachers that Harry has received a flying broom, something first-years aren't allowed to have; to his chagrin, he finds out all teachers have suspended the rule since Harry is now a Seeker.
When Harry asks Professor Dumbledore what he sees when he looks into the Mirror, Dumbledore answers that he sees himself holding some socks, objects he very much likes to receive, but people always give him books instead (Harry doubts whether Dumbledore is really truthful here).
A scene from the book was filmed but deleted from the movie: Harry finds a collector's card explaining that Dumbledore is famous for having defeated the notorious wizard Grindelwald, and his work on alchemy together with Nicholas Flamel, which prompts Hermione to find a book on Alchemy, and deduce that the secret object guarded by Fluffy is the Philosopher's Stone. The book features another Quidditch match, Gryffindor against Hufflepuff; as Snape is the referee, he predictably favours the latter team, but Harry wins the game by catching the Snitch five minutes into the game.
The subplot of Norbert the Dragon is more detailed. As breeding dragons is illegal and Draco has been spying around, Harry and Hermione convince Hagrid to let Norbert go off to Romania to live with others of his species. They use the Invisibility Cloak to carry Norbert to the Astronomy tower where some friends of Ron's brother Charlie (who works in Romania as a Dragon Keeper) picks him up during a transport. Malfoy tries to rat them out again, but as they are invisible and he is outside his dorm during curfew hours, he receives detention. In the exitement, Harry and Hermione forget the Cloak and are discovered; Neville is also caught when he wants to warn them. This leads to a major point deduction for Griffindor house and all three having to go with Hagrid to the Forbidden Forrest as punishment, along with Draco. In the movie, Ron gets punished in Neville's place, because in the book, Ron was in the Hospital Wing due to a dragon bite.
While in the forest, Harry starts out with Hagrid and Hermione, while Hagrid has his dog Fang look after Draco and Neville, and they meet several more centaurs, who seem to have an uneasy relationship with humans. When Draco starts pestering Neville again, Hagrid teams Draco up with Harry. In the film however, Harry is paired with Draco and Fang in the beginning, while Ron and Hermione are paired with Hagrid.
When Harry, Ron and Hermione are going after the stone, there are two more rooms after the Chess Room; one contains a Troll that has luckily been knocked out already; the other contains a Potion Riddle designed by Snape, comprised of 7 potions, and the door to the next room is blocked by magical fire. Hermione easily deduces which two potions must be drunk to pass through the fire, but opts to let Harry go and stay behind herself to care for the injured Ron. Due to this omission Hermione remains with the injured Ron in the Chess Room.
While in the hospital, Dumbledore agrees to answer Harry's questions, but he refuses to tell Harry just yet why Voldemort wants him dead; only that the sacrifice made by Harry's mother gives him strong protection against Dark Magic. He also admits that he gave Harry the Invisibility Cloak, which was given to him by Harry's father for safekeeping. Dumbledore also mentions that Harry's father once saved Snape's life, which Snape still resents and may be the reason why he hates Harry.
Everyone passes their exams for next year, with Hermione, naturally, on top of her class. Harry gets a notice stating he is not allowed to use magic outside school, but he chooses not to tell the Dursleys, so he can keep Dudley off of his back for a change.
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