Bows often have greater range and power than crossbows (depending on the type). The only reason that crossbows were used is that to use a longbow requires huge amounts of training from a young age to get the strength and technique right. Crossbows, on the other hand, require almost no skill, all you have to do is wind up the crossbow, put a bolt in, point, and shoot.
I'm going to have to ask for a source on this one...
The typical archer in a Medieval army was a conscripted farmer given a shortbow that was all but useless against armored targets, which was exactly how the knights of the era liked it.
The landed gentry who rode into battle on horseback in full armor hated crossbows because crossbows were actually capable of piercing full plate armor, unlike standard shortbows, which were only effective against the similarly conscripted leather-armored infantry.
Only a very few select types of bow were capable of competing with crossbows, and you're pretty much talking about Welsh longbows and maybe a few others at that point. Few others were willing to spend the time to actually have trained archers, most other European nations considered archery a peasant's job, and other cultures that did have nobles with bows were typically horse archer cultures who had to use shortbows by the necessity of being mounted.
A Welsh Longbow with a highly trained archer was capable of piercing plate mail from relatively close range, to the point where either you had to stand your archer right up to a charge, or you had to ambush knights from a forest or put them on elevated ground or behind a palisade that you somehow tricked a formation of knights into charging.
The main advantages of bows are their rate of fire.
Cheap crossbows aren't much better than bows, and are less accurate, but comparing a cheap crossbow with a conscript to one of the best bows ever made with an expert archer is a disingenuous exercise.
Windlass crossbows were capable of piercing armor from a much greater distance than a standard bow, but took so long to load that they were essentially worthless outside of something like siege warfare, where you could take all the time you wanted to reload when firing out of arrow slits, or from behind pavise shields up onto the battlements to try to snipe off castle defenders.
A stirrup crossbow, however, could still kill a knight in platemail armor at a range of 200 yards, which beats the range at which a longbow was deadly by quite a bit. A longbow was accurate out to a greater range, but by that range, it had lost so much power that it would no longer be capable of piercing the armor of opponents, and could only be used as raining volleys that might get a lucky eye-slit shot, or maybe kill an under-armored horse.