Medium/Normal/Regular is the baseline difficulty to enjoy a game, not too easy, not too hard. Too easy especially can take immersion away from a game and in some cases, completely kill the way the game is supposed to be played. A good example of this is the Hitman games. The higher the difficulty, the more you gonna have to become "Hitman" and think more tactically. Easy can turn it into a shooter.
Thing is, in a lot of games today "medium" is the new "easy". In Dead Space for example I barely died and only got killed by enemies that deal one hit kills. I was always 10 times stronger than anything in the game. The horror turned into action for me.
If you play on hard the first time, it's more rewarding and the game lasts longer
If you really want a challenge though, Resident Evil 4/5 and the last few CoD games can be a bitch on the hardest difficulty.