I can only speak for what I read in the Iliad and Odyssey, but these are my sketchy thoughts on what the movie made better and worse.
Things that the movie messed up:
a) Trivialising the Gods: The general audience probably wouldn’t have liked this, but the Gods explain a whole lot of things. The turn of the battle does not work without them, as was evidenced by the laughter from the audience when the Trojans retreated for no reason. It would have been neat (but a strain on the CGI budget) to see the likes of Athene racing down from Mt. Olympus to whip Odysseus out of trouble. As would the whole interplay between Zeus and Hera, and the god’s reaction to the Greek wall.
b) Where was Diomedes? Teucer? The Ajaxes? I suspect the guy with the mace was supposed to be one of the Ajax’s, but is he referred to be name? Diomedes and Teucer have huge roles in maintaining the battle while Achilles is absent, and though I realise that the film telescoped the whole affair, they deserve a mention at least. Diomedes is practically Hector’s Greek equivalent for a good part of the story.
c) Killing Melenaus and Agamemnon. You don’t just kill these guys… I admit that Homer’s reluctance to kill any of his heroes was a bit extreme, but why not pick somebody who doesn’t have a role in the Odyssey? At any rate, Agamemnon’s real fate is much more appropriate.
d) The importance of being remembered: The Iliad did not present this message anywhere near as prominently, and its addition changed Achilles from someone who can’t control his emotions to an arrogant fame seeker.
Things the movie improved on:
a) Speeding things up. In the Iliad, they’ve been waiting outside the city for 10 years before they attack. This is plainly ridiculous, and I don’t think Homer even found a convincing explanation for it. Maybe the movie made the attack a little too rapid, but it makes for a better story.
b) No river chasing Achilles. This really did not work in the Iliad, and it was another problem that was feebly solved by a God. Good riddance.
c) The funeral games: Again, this added nothing to the story of the Iliad whatsoever other than giving the heroes another opportunity to congratulate themselves.
d) The love stories. The Iliad is a bit too barren when it comes to relationships. Helen appears to have no say in her predicament whatsoever, and it actually seems she was kidnapped. It was nice to give her a voice, and Achilles’ girl (forgot the name) did a good job of filling the void left by the gods. The romance of course, also makes for a much fuller ‘epic’.
e) The death of Hector. The way Achilles does this in the book is pathetic. A goddess, Athene, comes down and tricks Hector so that Achilles may kill him. The fight in front of Ilium’s walls is better representative of their characters, and it seems like something Homer would do.
f) Paris running away from Melenaus was far better for his character development than having Apollo whisk him to safety.
g) The line “That is why nobody will remember you.”. That was classic – way to crush a kid’s dreams, Achilles.
I apologise for anything that I’ve got wrong, for my attention did have a habit of slipping during both the book and movie.