As it happens, u/wilymaker won one of the Best of April posts on just this matter entirely! And deservedly so. To round out their excellent answer, this is also an FAQ; this link should take you to the appropriate section, Transition from Bows to Firearms.

It's always important to frame the question - is this on an individual level or on a battlefield scale? Are we only judging the results from a one-on-one skirmish or factoring the logistics of mass manufacture in an industrial era versus the artisan-level production of bows and arrows?

Many of the perceived advantages of bows, even from historical perspectives by later writers, assume the best of the bow in the best of circumstances while they themselves are in less desirable conditions. Realistically, we should remember that any culture that encountered firearms adopted them rapidly over their traditional equipment. The Japanese, for example, were said to have manufactured more firearms than Europe despite the image of the samurai being a traditional warrior with bow, spear and sword.

Let's go through each of the perceived advantages.

Bows are more accurate

The inaccuracy of muskets is greatly exaggerated. Doctrine largely dictated engagement range rather than accuracy. Broadly, a musket is predictable to around 50 metres, beyond which there was no point in aiming at a single target. This is significantly better than a bow. A skilled archer could consistently hit a 50m target, but this required a respectable amount of skill and training. What's really telling with this comparison is that accuracy for archery depends on the person. For muskets, aside from flinching, the weapon itself determines accuracy. If the archer is unable to shoot well, either through lack of training or through injury, they are ineffective.

The French General de Marbot, in fighting against the Russians, wrote of his encounter with irregular Baskir horse archers:

With much shouting, these barbarians rapidly surrounded our squadrons, against which they launched thousands of arrows, which did very little damage because the Baskirs, being entirely irregulars, do not know how to form up in ranks and they go about in a mob like a flock of sheep, with the result that the riders cannot shoot horizontally without wounding or killing their comrades who are in front of them, but shoot their arrows into the air to describe an arc which will allow them to descend on the enemy. But as this system does not permit any accurate aim, nine-tenths of the arrows miss their target, and those that do arrive have used up in their ascent the impulse given to them by the bow, and fall only under their own weight, which is very small, so that they do not as a rule inflict any serious injuries. In fact, the Baskirs, having no other arms, are undoubtedly the world's least dangerous troops.

However, since they attacked us in swarms, and the more one killed of these wasps, the more seemed to arrive, the huge number of arrows which they discharged into the air of necessity caused a few dangerous wounds. Thus, one of my finest N.C.O.s. by the name of Meslin had his body pierced by an arrow which entered his chest and emerged at his back. The brave fellow, taking two hands, broke the arrow and pulled out the remaining part, but this did not save him, for he died a few moments later. This is the only example which I can remember of death being caused by a Baskir arrow, but I had several men and horses hit, and was myself wounded by this ridiculous weapon.

Marbot might be writing dismissively of irregular cavalry archers, but we should consider that they are fighting against the ideal target - stationary massed unarmoured infantry. To shower arrows onto this formation and only achieve a 10% hit rate with little effect indicates that the accuracy of archers may be overstated and overvalued, at least in the gunpowder era.

I think it's fair to say that in battlefield conditions, the distance at which a skilled archer could reliably hit a target would not be that much more - if not less - than an average soldier with a musket - but requires far more effort and training.

Bows are quiet

This is undeniable. One weapon ignites powder that goes BANG, the other doesn't. This isn't exactly advantageous in a battlefield context, however.

Bows are quicker to reload

This is true for the most part. Benjamin Franklin, in writing to Charles Lee in 1776, praised the bow for its ability to "discharge four arrows in the time of discharging and charging one bullet", among other perceived advantages.

There's an element of "so what?" to this. The main limiting factor of the bow by the end of the 16th century was the common use of armour. Armour had reached the point where arrows from the most powerful bows - which in turn required the most long-term training - were increasingly unable to defeat. The reduction in the use of armour came about as firearms became more common and reliable, by which point the bow was no longer in regular use.

While a bow can be shot quicker, if it is shot with less accuracy and with less effect but requires more effort from a smaller number of soldiers, it's easy to see why the musket was adopted as the weapon of choice for larger gunpowder-era armies. One thing to consider is that the musket didn't just replace the bow - it replaced the spear. The bow was always a supportive weapon in an army - the musket was the main weapon. Initially they were protected by pike formations, and with improvements in muskets, the pike too was phased out in favour of having more guns.

Arrows are reusable

Yes...but seldom will this be applicable in battle.

But since we're on this note, we must consider that muskets, powder and shot were produced on an industrial scale. Bows and arrows were made by hand and demanded more materials at greater cost.

Remember the scale of the battles in the context they were fought. The great "archer" battles of history, such as Crecy and Agincourt, had perhaps 5000-7000 archers. An army during the Napoleonic Wars might be 50,000 to 100,000 strong.

Conclusion

Donald Featherstone, in The Bowmen of England, perhaps summed up the end of the bow well: the gun is a weapon that can be perfected. Time has shown us this - improvements in reliability, rate of fire, accuracy and power. The bow's weakness is locked to the person who is using it.