the only thing I can add to the excellent things stormerne said is that there is a limit to how much ground troops you can stuff into an area and use to attack an enemy, while you can put an almost unlimited amount of bombers in the same place.
yes, of course you can defeat the AI without planes, even at deity, even with a smaller and technologically backwards army, because AI is very bad at using units. I once repelled an information era invasion force, killing an helicopter army and modern armor corps, with two crossbows and a handful of swords. I don't have experience here, but I'd surmise they are not useful against humans either; the one time I tried multiplayer, in half an hour I can never stay in a game more than five minutes before everybody disconnected/quitted, so I don't think human games tech flight that often... regardless, planes are strong, and after you pay the initial price of the aerodromes (which provide very useful transportation anyway), they aren't particularly expensive.
EDIT: on the other hand, it is worth remarking that since the AI is at the moment incapable of defending against them, using airplanes feels like shooting on the red cross, even at deity, and it makes the game too easy for my tastes.