I've completed the game on hard and played around with other difficulties.
The only differences appear to be:
Quick time events are, er, quicker. You have less time to hit that one button.
Damage done to you by enemies is higher - on hard a single rifle burst can kill you.
Non head-shot damage appears to be lower.
On my hard playthough I fully upgraded all the guns and skills just before the point of no return (the game warns you when you reach it), but that was with all treasures found and all challenges complete.
On Xbox there don't appear to be any achievements related to difficulty.