Hardcore Gamer~
A gamer who takes games will less of a casual approach. A Hardcore Gamer is not so much singled out in his choice of game, but how he responds to them, a hardcore gamer strives to master the game, to improve in his skill, to play competitively. He buys games to get good in them, in addition to having fun. A hardcore gamer cares about his reputation as a player, he desires to make a name for himself in someway, on some scale, either through his friends or on a larger scale of people/acquaintance. A hardcore gamer (with exceptions) form or join clans with friends (off or online).
There aren't many single player games that a Hardcore gamer will immerse themselves in on a large time scale. They play for the competition of it.
Casual Gamer~
A gamer who plays games solely for the entertainment factor, his desires to improve at the game are not that of a competitive scale of things, but rather to know how the mechanics can best suit him to complete the game. He cares not about or even acknowledges any reputation he builds for himself. Competitive play is not used in the same context here, yes he strives to win, but not to reign as best. Casual gamers are mostly fairly satisfied with the single player experience, although they may get into competition it is part time and not taken as seriously as a hardcore gamer would.
The primary difference isn't the games they play, but how the game reflects upon their aim, a casual gamer can play Gears of War 2, and simply be content with the campaign or horde, and a hardcore gamer can get into Sonic Unleashed, but wouldn't garner as much satisfaction as he would in a competitive environment.
That's my two cents.
This is an old and tired subject.