A serious gamer is someone who devotes a good bit of their life to gaming. Usually they go on to be player testers, game creators and things of the like, or are on a devoted team that travels the country competing in tournaments and leagues and make a living of gaming.
Then you have your devoted gamers. Ones that don't pursue trying to make money from gaming, but just love to game. These are the ones who spend 40+ hours a week gaming, and then when the weekend comes, they aren't going out to the clubs, but rather having lan parties at their house througout the weekend. Or no lan party, and just dedicating their weekend to gaming alone, or online on multiplayer games.
A casual gamer is someone who doesn't spend a lot of time or money into the gaming industry. They will get addicted to one game and most likely just play it when they can, basically when there is nothing else to do and they want to pass some time.
This is basically how I categorize gamers, and there are other types and some may vary, just depending on how mixed their feeling are towards gaming, and how much of their free time they want to spend on gaming, or how much free time they want to make for themselves so they can game.
Like myself, I would consider myself a devoted gamer because I am disabled and unable to work so I am at home 7 days a week unless I decide to hit the clubs a night or two throughout the week or weekend. Other then that, you can find me sitting behind a computer or console, putting away countless hours of gaming.