Crossbows were generally more powerful at a short range, but due to some science we don't have in front of us right now (wind resistance stuff, lots of mechanics), they're not as effective at a long range. Longbows fire faster and further than crossbows, but they have less stopping power. I.e. they can't penetrate armor quite as well.

Also, an expert longbowman takes a decade or more to train, while a crossbowman only takes a few months to train.

In essence for crossbow vs. longbow:
[b]Crossbow:[/b] punches harder - shorter range - slow to reload
[b]Longbow:[/b] smaller punch - longer range - quick to reload