One reason I usually prefer mines over windmills is because of the chance to strike it rich. I've found that it seems to increase my chances of having a resource I've just researched if I have mines on all my hills. Not sure if that's proven or not though.
Probably just luck, or maybe the resource otherwise pops up under a windmill and you never notice it. There is a tiny chance per turn of each mine "discovering" a new resource deposit, but that's never happened for me and I usually go almost all mines too. Relying on flat tiles for resource production screws up your food (workshops add -1 food, windmills don't get the Biology +1 bonus farms do). On the other hand, windmills do add cash with certain technologies and watermills are always cool, especially for Financial civs.