And don't forget all the fun of limited slip differentials or locking differentials. So a 4wd vehicle, like my Wrangler, is actually one wheel drive under normal circumstances. The engine sends power to one rear wheel. If you have a limited slip differential and that one tire loses traction, the axles starts sending power to the other rear wheel. If you don't have one and you get stuck in the mud, this is when you see one wheel spinning in the mud and the other doing nothing (No LSD in that case.)
In 4wd you actually go up to 2 wheels powering the Jeep, one on the rear axle and one on the front. If you actually have locking differentials then you can "lock" them into place with a switch and effectively send equal power to each wheel on the axle.
None of these takes into consideration electronic traction control, which uses your brakes to slow one wheel down while forcing the other to spin. It's all so simple and clear!
Seriously though, for your situation either would be fine, it's just that 4WD would require you to choose to put the vehicle in 4WD, while AWD takes care of it all for you. As mentioned before, AWD typically runs biased to FWD and shifts power to the rear axle as needed.