Some of the missions you go on will have a different path to choose from. The overall outcome is the same but you get to explore a different level and new cutscenes. I think this happens once a chapter? Not sure.

The green exclamation points you completed at school on your first playthrough will now be green checkmarks, you can watch them again at no extra charge to your time. Also, quests that weren't done the first time (such as seeing Emina strip) will have progress carry over.

And seeing as how your levels carry over, completing events of higher difficulty will give you better rewards.

There are a ton of things I couldn't do on the first playthrough because you needed to be level 50+ to do them. The second playthrough was more rewarding for me.