The difference between bows and crossbows is as following (and if anyone cares, I'll explain them in comparison to guns, too.):
Bows require a lot of strength to draw, and even more to keep them drawn - which makes careful aiming very hard. Also, they're a lot more difficult simply to use at all, due to the high tension in one's hand - ears and fingers are at high risk.
Crossbows are complex machines to build, but simple ones to use. Basically, putting a bow on a stick takes most of the more difficult actions away from the firing and reloading procedure. A crossbow of a strength equal to that of a bow is *almost* as fast on the reload, since it takes no longer to draw it, and the only extra action is placing the bolt in the sled. Aiming a crossbow is also easier, since you can take your time without having to keep it at high tension. And a further advantage is that you can use a pre-loaded crossbow straight away, without having to notch and draw first. These two bonus perks make it extremely useful in sieges, both defensively and offensively, since you can snipe defenders off the walls (offensive), and/ or hide behind cover for reloading (defensive).
All that goes for crossbows no stronger than bows. However, once upon a time, some crafty murderer realised he could actually increase the draw strength of the crossbow far beyond what one might be able to draw with an arm's strength. So all manners of machinery were invented to make it possible to draw those beasts, which then also achieved far greater range and power than bows, at the cost of it taking a lot longer to draw the crossbow.
And we all know how far you can take a crossbows strength...the arbalests had ranges up to almost a kilometre, and roman ballistas were field artillery based on what's basically a hugely oversized crossbow.
So, in summary, bows are simpler to make and always at least a little faster to shoot with, but crossbows are simpler to use and can be upscaled to practically any strength, range and accuracy (at a cost to manufacturing complexity and reload times).