Well, I did a bit of research, though that didn't mean this is entirely accurate as I don't know how knowledge the people providing details are.
With a compound bow vs compound crossbow, the crossbow will have better range. However, old style English longbows could have accurate range at a greater distance than a one person handheld crossbow but that required someone with a lot of skill with bows. And accuracy and range decrease with more shots as you get tired of using such a strong draw.
As a note, I can't be sure the comparisons for English longbows is comparing against modern crossbows, so that may not be accurate anymore.
Anyhow, in terms of gameplay, unless crossbows are considered to be a higher tier bow instead of a separate weapon type, I don't think they should be all out better even if they realistically are. Each should have some benefits or there is no reason to have bows at all.