As other members mentioned it depends on a lot of things, the first and most important one is which faction your are playing with, for example if you choose Milan, their Genoses Crossbowman are extremely powerful and they can recruit them early in the campaign. See below the ups and downs of each:

Archers:

Some may deploy stakes (England higher end archers, Turks Jannissary archers, etc...)
They can use falming missles
They shoot faster then Crossbowmen

Crossbowmen:

They have Piercing Armour bonus, this means in late campaign they are stronger then archers
They are better in melee then most archers
They have a better defense as they have the shield bonus