The thing I like about Hard is that it can force you as a player to take a better look at all the mechanics the game presents to you and figure out both why it exist in the game, but also get creative with how to use it in not just scenarios that you have been taught by the game to use them in but also in instances where you've been pushed to the edge and had to come up with brilliant solutions on the fly from your knowledge of the mechanics that you might not be pressured enough on Normal difficulty to think about.
Depending on how difficulty is implemented it can also sometimes help with immersion, by making the game a bit more fair for, well, the game. Like, perhaps I shouldn't be allowed to walk through 5 fireballs before I die? If that skeletons that's hitting me actually managed to overcome the ultra loaded murder kit that has been provided to me as a player and broke through X amount of layers of defenses from decked out gear, perhaps it's OK to give me the player a whole bunch of damage that I should be wary of allowing to happen again, since this is clearly one badass skeleton and should be treated as such :)