Based on this, there are a few things you can extrapolate as the difficulty level increases:
Skills based on damage increase more slowly, and in turn, the slower you level up (thanks yx.)
Followers can be knocked out, and in turn, killed, more easily
Your effective health decreases
Enemies' effective health increases
Possible note: Weapon Skills aren't based on final damage; they are based on base damage of the weapon, which is unaffected by difficulty. Increasing the difficulty means that you have to hit each enemy more times, which speeds up skill growth. This is very noticeable when you increase the difficulty as my one-handed increased about 6 levels after 2 dungeons.
Nope, as far as i know difficulty is handled in this game like it is in most elder scrolls games and that is to say it doesn't effect anything except... well difficulty. :)
You can only train 5 skill levels per (character) level.
So if you are level 10 and train destruction at a trainer you can train 5 times. Then you have to get to level 11 to be able to buy skill levels at a trainer.
That is the only restriction I know of (being lvl 62 atm).
Best Answer
According to the UESP, the difficulty level affects the damage you deal and the damage you receive:
Based on this, there are a few things you can extrapolate as the difficulty level increases:
Possible note: Weapon Skills aren't based on final damage; they are based on base damage of the weapon, which is unaffected by difficulty. Increasing the difficulty means that you have to hit each enemy more times, which speeds up skill growth. This is very noticeable when you increase the difficulty as my one-handed increased about 6 levels after 2 dungeons.