How early are we talking?

14th century small arms were terror weapons more than anything.

But by the 15th century, early guns did offer advantages over bows and crossbows, albeit not necessarily decisive ones. For one, they had a flatter ballistic trajectory than either - meaning aiming is much easier. Lethality means little if you don't hit the target, after all. Troops could also carry more ammunition, as a lead ball and powder take up much less room than even a crossbow bolt, and lead balls, for that matter, could be cast by even laymen with a few basic tools (since lead has a low melting point). And bullets had superior penetrative power in comparison to most non-English bows and probably even the majority of crossbows - important in a period when metal armor was widespread. On top of all of that, firearms require very little in the way of physical attributes to use.

Basically, you get a weapon that's as deadly as a crossbow, but easier to use, able to sustain fire for a longer period during battle, better against professionally-equipped troops, and terrifying to boot.

Every development from that point on increased the advantages of firearms, and decreased the disadvantages, until firearms were no longer 'competitive' with bows and crossbows, but explicitly out-competing them for military purposes.