Using armies of massed archers was very labor intensive and expensive. You can often train someone to use gunpowder weapons in a few hours, but training someone like an English longbowman would take years to develop the skill and musculature required to draw the bows of the age. One of the complaints against crossbows, however, was actually how unsporting it was for an untrained peasant to be able to kill an armored knight from long distance, and they were actually banned for use against other Christians under the 2nd Lateran Council under Pope Innocent II, but there are significant doubts about how effective this was in curbing their use, and the ban is arguably in itself a testament to their effectiveness in combat.
Besides the actual archer you also have to figure in the difficulty in making successful bows or crossbows in an age before advanced engineering and metallurgy had taken hold, and the limitations of bows and crossbows. Bows and crossbows ranged from relatively unpredictable to useless depending on the weather conditions, which in Western Europe are often wet and cold for periods of time. Crossbows, bows, and Crossbow bolts and arrows are also a significant expense for what may be a one time use. You could not reliably expect to pick up previously used arrows on a battlefield and expect them to function like the ones that you had been practicing with prior to the battle, which could mean your shot misses and that heavily armored knight you were aiming at closes the distance and cuts you down.
Speaking of the armor, this was often specifically designed to negate the effects of crossbow bolts and arrows. A bullet or musket ball (or cannonball) can reach you much quicker and with more force than an arrow or crossbow bolt from the middle ages. Body armor may never really have disappeared from Europe, but without a doubt it became less about full coverage and more about protecting your vitals, as it was soon realized that you were never going to be effectively safe from a bullet.
Gunpowder also didn't completely take over for a very long time into their development. It took many failed bayonet charges to fully appreciate the potential of firearm use, but before that firearms were just one component in militaries of the day. Guns, bows, and crossbows still share the same weakness, a limited number of times that the weapon can be used before it's effectively a club, and guns come out on top here because it's much easier to fix a bayonet to a long rigid object and use it effectively than it is to a rather bendy bow or a short and awkward crossbow, and military leaders of the gunpowder age eventually came to the conclusion that they could reserve expensive swords for the cavalry and officer corps if the average foot soldier is going to be nearly as effective in a melee with a gunpowder weapon.
Great question.
I'm curious about why crossbows didn't completely replace bows in Europe? From my limited understanding cross bows were vastly preferred to bows in Chinese armies (ignoring steppe archer troops). Just how common were crossbows compared to bows in different European armies/time periods?
The premise of the question is slightly off.
Neither bows/crossbows nor early firearms were good enough to protect themselves and prevent close-quarter combat, especially a cavalry charge.
In fact in the late 16th century the Netherlands were complaining they could not get enough men to be pikemen because too many opted for making useless noise and smoke by using an arquebus. See this thread, courtesy of /u/Polybios.
What made the musket replace pikes was the bayonet. The bayonet combined the gun and the spear into one.
It took a quite a while for firearms to really "completely take over" rather than being combined with pikes and other melee weapons, but in the 16th century, when the matchlock really became widespread, mercenary Humfrey Barwick did note that battles seemed to be far more often fought at range rather than the two sides closing to melee like they did in the past. Barwick gave a number of reasons for why he thought firearms were far superior to bows, but the big one which really impressed every firearm proponent tended to be the massive force of the weapon. A warbow drawn by an extremely strong archer might shoot an arrow with up to 100-150 Joules of kinetic energy, but firearms relying on stored chemical energy could achieve far more. In tests performed at Graz Austria, one antique 1580s musket was able to achieve a muzzle energy of nearly 7000J. According to Barwick, armored soldiers needed only to tilt their head forward to protect their face and they could advance into a hail of arrows completely unafraid, but heavy muskets on the other hand though could pierce good quality armor as far out as 200 paces.
More power = better also tended to be the prevailing opinion when comparing different firearms. Sir Rodger williams wrote about calivers (a firearm which was a bit lighter and had a smaller caliber than the musket): "the Caliuers may say they will discharge two shot for one, but cannot de∣nie; but one Musket shot doth more hurt than two Ca∣liuers shot". Barwick also mentioned that he thought the musket to be much better than the caliver/arquebus. Robert Barret, another writer from this period wrote that while normally the ideal ratio of shot to pike in a company was 1:1, if the shot could be armed entirely with Spanish muskets the ideal ratio changed to 2 shot for every 1 pike.