C-h Freese:
Technically Gamer is far more generalized.. as chess and poker are games and can count as Gamers.
To me a gamer
plays* games, usually a broad swath of types, I play roleplaying games, board games, video games, so I call myself a gamer.
The person who only plays one type, or one game isn't a gamer. Though it's pretty rare to find people who only play one game outside the Boomer and older generations.
* And buy this I mean they chose that as their primary hobby or predominant mode of leisure activity. Someone who only plays games once a week, for four hours
isn't much of a gamer, even if they play a broad swath of different over the course of a year or two (mostly because they are probably choosing other leisure activities for the most part).
To me a roleplayer is someone who plays rpgs, generally only tabletop, but I know people who consider MUDs and MMOs sufficiently immersive that they count playing those games as roleplaying. I don't, just as I wouldn't count reading/playing a choose your own adventure book as roleplaying, even though in some you very much build a character and play a role. They might also be a gamer, but then gamer is a broader category into which roleplayers can (but do not always) fall.
Likewise I don't count playing Skyrim as roleplaying, despite it's "rpg" status.