From what I've heard basically you just take more damage from mobs, not as many mobs spawn, and probably more damage from falling. Other stuff is unchanged like dig speed etc. Anything that deals damage I believe does more on hard. It isn't really all that different.
I also would like to know specifics in general if anyone has done any extensive testing. I actually wouldn't mind LESS mobs. I literally have too many on normal. Went down into an epic cave system that was basically infinite and had like 100+ mobs in there. Had to peaceful to get rid of them. Would of taken so many swords to kill them all.
Dig speeds and such are the same.
You take double damage from mobs (meaning 4 damage a hit, aka you'll get 3 shotted without armor).
And there is a lot of controversy about mob spawns right now, but clean room testing shows its all the same. (I just know there will be tons of posts after me arguing this, but the argument is moot until someone does legitimate testing proving otherwise.
As for the other various damages, I'm not quiet sure since I've never tested those specifically.