Hardcore Gamer: A person who cares about their games way too much. Enjoys labeling games that aren't made for the top 1% of most-skilled gamers as "casual" and for kids. Is probably enraged whenever people who don't make video games their primary hobby dare to play them anyway, and that companies like Nintendo make games for those people. May actually sit in front of the TV/computer playing a certain game enough to be one of the top-ranked people on a game, instead of doing important things like eating healthy food, getting sunlight, bathing, dressing decently, and interacting with other human beings. Scoffs at people who dare play games any way but the absolute most effective way. Scoffs at people who tend to play at difficulty levels below Ultra Hard Masochism This Is PAIN. Will play most kinds of games, but tends to favor "strategic" genres like FPSes, RPGs, MMORPGs, and RTSes.
Casual Gamer: Someone who plays games, but doesn't make it their entire life(ie. they play less than 40 hours a week). Can still know quite a bit about games, but isn't an arse about it. Plays games to have fun, not to be the best person in the entire world at Generic FPS 9000. Can actually get pretty good at a game, and is willing to play games with some challenge, but won't bother with Insane Masochist You-Hate-Yourself-And-The-Game-Does-Too difficulty levels and won't put in the time required to be *as* good as a hardcore gamer. Will play most kinds of games, but tends to favor those that are fun to play. May have actually seen sunlight in any given 24 hour time period.
In short, they're the same damn thing, it's just that casual gamers don't make gaming their reason for being and don't get offended when other people dare to enjoy it in a different way than they do. They also tend to care more about what's fun than about what fits some arbitrary standard of challenge or strategy(of course, truly fun games do have both of those). Of course, hardcore gamers can't comprehend playing games to have fun instead of to be the absolute best at them, but I guess they tend to take their games as seriously as a job anyway.