To put it simply, without trying to cross over any boarders. Gamefowl are breeds that were bred for cockfighting a long time ago and still today in some countries though is purely exhibition (show) in a lot of countries like the US now. Game is more to describe their nature (where roosters will fight other roosters, and sometimes hens fight hens) than breed really, but old english game bantams, modern game bantams, and breeds like that hold the title without being game because they were derived from games.. You should look into the birds, i haven't found a game yet i didn't like. Sumatras, and many others were once games and games are in many of the histories that created Rhode island reds and other breeds.
ETA: Oh yeah, and then you have game, like gamebirds (Quail, pheasant, Chukars) which are called game because people hunt them.
Thank you. I have two chickens that i was told were game. They just are built different from the other ones, but they lay the biggest eggs of the bunch.