At least in AH speak.
Player would be a person who plays the game, be it strat, furballing and plays for the sake of flying the aircraft. They might score high, but not a priority as gameplay is. It is a short definition, because it is a general term, refering to many different types of players. Thet are generally a good part of the community, both in numbers and personalilty.
Gamers seek out other ways such as manipulating score to give themselves high ranking with little effort, pork' n' auger to drop a target saving themselves time by not having to fly home. Repeatedly suiciding a CV would be gaming. Flying to vulch without trying to take the field, gaming. Spawn camping, gaming. Basically they are trying to game the game in any way to give themselves an advantage, without actually playing the game.
Generally players tend to see gamers as a bane of existence in the virtual skies of AH. They tend to be younger in age, though not always. Generally not interested in the history of aircraft. Generally not good for the community as they add little or nothing to gameply, immersion, or community.
So, in the real world, I'm a gamer.
In AH, gamer is a bad word and wouldn't want to be called one.