1. Ron and Hermione always fight when they are with each other. You don't see very much of this. In fact the only argument I recall them having is about Scabbers and Crookshanks.
2. They don't spend much time in the Divination room. So you don't get why Hermione doesn't like Treelawney.
3. The one part I was waiting to see... Harry is playing Quiditch and Malfoy and his friends dress up like Dementors and try and scare Harry. Harry uses the Patronus Charm on them and they are the ones that end up scared. It was one of the best parts in the book.
4. A lot of parts were left for people to work out the pieces. It isn't explained who made the Marauders Map you have to deduct the info for yourself. Same with the relationship between Snape and Lupin. Snape didn't like Harry's father; Lupin was one Harry's father's best friends.
5. Someone mentions a potion to Lupin before he turns into a Werewolf. This explains the reason Snape was following Lupin into Shrieking Shack. It's actually supposed to make Lupin calmer when he is a werewolf or something like that.
6. Harry gets his Firebolt at Christmas time. This is important for the Quiditch cup. Which is another semi-important thing in the book. It's the team captain's last year at school. Gryfindor hasn't won the cup in a long time. Harry on his Firebolt = unstoppable.
7. There was no going away feast or scene at the train station. While probably not that important it certainly would have made for a much better ending.