Some of these things have already been mentioned:
Hamilton and Bates are cold; Davidson is not as cold.
Davidson is D1 in some sports; Hamilton & Bates are NESCAC, D3.
Davidson is like a suburb to Charlotte, Lewiston is a small city (the largest of the three), and Hamilton is very rural.
Bates is the most liberal overall, Hamilton is fairly moderate for a LAC (probably center-left overall), and Davidson is the most conservative of the three and probably fairly moderate overall. But lively discussion can be found at all three, or avoided; you can find kids with interests similar to yours at all three; and none of them is known as a particularly militant setting politically. In other words, I wouldn’t put too much into the politics of these schools. You coukd be yourself and get along just fine.
Most importantly, are you a Continental, a Bobcat or a Wildcat? I personally feel like “wildcat” is a mascot cop-out: can’t you pick a cat species? All the non-domesticated cats are technically wildcats. 
If it came down to the mascots, I’d be a Continental.
Red and white, maroon and white, or buff and blue?
They’re all good. I’d choose on academics (look at majors, academic schedules, curricular styles at least), sports/social vibe, location/environment, and cost if they are not similar.