What holds up well is, first and foremost, of course, the story. But if we accept a good story as a given, the next thing that holds up is what, for lack of a better word, I'll call nuance: the suggestion of depth; the viewer's ability to "discover" things every viewing: quips, excerpts of dialog, visual jokes, visual extravagances or clues. I confess--and may all Harry Potter fans forgive me--to truly disliking the first Harry Potter film. Yes, yes, the characters looked right, and the scenery looked right and it kept to the book, but it was dead, one-dimensional, like watching a film strip. The second Harry Potter film was far better, but the third shines. It has depth, being built on several layers of nuance.
Here are some of my favorites (not in order):
The cute exchange on the stairs when the Fat Lady has disappeared, and Ron says, "Maybe Neville forgot the password," and Neville says, "I'm right here," and Ron says, "Oh, sorry."
The Fat Lady herself who is played by the excellent, and very funny, Dawn French.
Harry Potter and Sirius fainting in the same way down by the lake.
The zoom-in shot on Lupin's changing eyeball.
The outside scenery--wow, wow, wow.
The way in which Ron, Hermione and Harry comfort each other when Buckbeak "dies" (the faint but elegant suggestion of beginning adolescent sexuality is quite effective).
The big band music used by Lupin. (Okay, okay, I like big band music.)
Malfoy's longer, blond hair (much more effective--and makes him look surprisingly older).
The startlingly improved acting by Daniel R. (especially in the beginning and the end).
The utterly amazing scene in the shack when four great British actors get to face off and totally steal the show.
The remarkable Gary Oldman.
The new Dumbledore (sorry, but I do like him better).
Robert Hardy--well, he's so very, very wonderful.
The Shrunken Head in the bus--that entire scene in fact.
Robert Hardy's sidekick at the Leaky Cauldron and the way he keeps offering Harry Potter food when Harry first arrives.
The first night in the dormitory when the boys are exchanging animal-changing beans--one of the great things about this film is how real the teens are. They act like teens, they talk like teens, they dress like teens.
Which brings us to the incredibly dirty pink sweater of Hermione at the end of the movie. I approve.
The speech by Lupin about Harry's mother.
The use of the stag down by the lake. This is a book reference and is actually one of the failings of the movie. It would have been very easy for Lupin to have explained to Harry at some point that his father was "Prongs" (of the Marauder's Map) and that his father used to change into a stag in order to help Lupin with his werewolvetry. As it is, the fact that Harry's Patronus spell appears as a stag (and that Harry would make the instant connection to his father) is utterly mysterious unless you've read the book. But it's a nice touch.
Harry's comment, "Professor Lupin is having a really tough night."