The big things are the UI and the big neutral objectives.
On blue side the UI, with your skills and minimap, is on the bottom, which ends up covering part of the screen that is not normally important to you.
On red side the UI is also on the bottom, which covers the direction your opponent is normally coming from. It results in situations where, for a split second at least, your opponent can see you but you can't see them.
Dragon and Baron also have small differences in the way their area is set up. Red team has a more natural path to the opening of Dragon pit, giving them an open area to fight in if blue team is contesting it. This allows blue team to control the area a bit better if they get there first, but also makes it harder for them to escape if the fight goes wrong.
Blue team, on the other hand, has easier access to the wall of Dragon pit. This gives them the upper hand when trying to steal, and if done properly blue team can collapse on red team easier if red team is contesting dragon.
All of this also applies to Baron, although the team colors are switched. Dragon and Baron have historically been early and late game objectives, respectively, which gives one team an easier time securing objectives in the early game, and one team an easier time in the late game. Riot is attempting to fix this by adjusting what sort of benefit Dragon gives, and adding an early game objective (Rift Herald) to Baron pit. Needless to say it's still a work in progress.
