Several reasons.
1: A crossbow was a peasant's weapon. Not because it was cheap, but because you could teach someone to use a crossbow in an afternoon and have them useful on a battlefield if only to upset opposing light troops. The joke that to train an archer you start by training his grandfather is only half a joke.
2: You might be able to load and fire a crossbow in 6-10 seconds using a goat's foot if you were hurrying - but longbows were much faster. (The 3e move action to reload for a light crossbow isn't far off).
3: Because of 1 the average skill of a combat longbowman was massively greater. Because of 2 the use you could get from a high skill longbow was much greater. So if you were going to specialise you went for the bow.
4: The crossbow was basically a European weapon of the late middle ages (yes, I know the Chinese even had repeating crossbows). The bow was used by masses of feared armies from English and Welsh Freelances to the Mongols, the Persians ("Ride a horse, shoot a bow, and speak the truth"), the Parthians (Carrhae, anyone?) and many, many other armies.
But basically it boils down to the crossbow being a weapon for relatively rich peasants and the way it changed the battlefield being to allow them to narrow the gap with professionals.
1. And most soldiers during the middle ages were peasants.
2. Archers can shoot arrows in rapid succession by only utilizing a fraction of the bows potential draw strength, but when fully drawing a longbow the rate of fire dropped and also tired out the archers rather fast. Thats why most archers in battle did not even fire at their maximum ability unless absolutely necessary. Also, aiming was harder has the archer could only hold the draw for a few seconds before his arm started to fatigue. Not a problem for volley fire, but aiming on a chaotic battlefield was only possible at a short window.
4. Also a common misconception. The romans already used crossbows and they never fell out of use by the European powers either, both on the battlefield and for hunting. Already in the High Middle Age in the 13th century armies tended to field more crossbows than bows.
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