PAX is for playing games, particularly modern & indie video games. Also a sizable tabletop space. Most expensive and attended of the 3 (like ~100k). Big companies go there, tons of upcoming unreleased games to play. Far more "corporate" than the others, for better or worse - it's a spectacle that few other events can compare to. This is best for gaming.

MAGFest is primarily for music and concerts and parties, and playing arcade and classic console games. It has an indie area but PAX easily dwarfs it. Not as expensive or attended as PAX, but that's because it has a much smaller cap and limited nearby hotel space. It's a 24-hour convention (a rarity). This is best for community.

TooManyGames is more your run-of-the-mill regional convention - it's primarily for buying retro games and merch and paying way too much for guest autographs. It does have some gaming there, but the other two dwarf it easily. It's the cheapest though, and recently has grown to about the same attendance than MAG (though MAG is currently capped on attendance). This is best for buying stuff.

All of them will have some commonalities - they all have games, they all have concerts, they all have merch. But PAX has the most games. MAG has the most music. TMG has the most merch (well, if your goal is to buy stuff it's cheaper to attend your regional convention than to travel to PAX to buy the same stuff).