Just a clarifying question, are you curious as to why gunpowder weapons developed given the superiority of other technologies or why did gunpowder weapons replace longbows/crossbows? The latter answer is a complicated one with more than a little speculation involved in it while the former is a lengthy-ish history lesson. Anyway, if you let me know I'll do my best to provide whichever answer you're looking for!

hi! not discouraging anyone from chipping in with more info, but you'll find several threads on this in the FAQ (link on sidebar):

Transition from Bows to Firearms

Im going to quote a bit from Russell Weigley, who wrote briefly on the subject. I think Weigley's quote really answers this question well, he is a highly respected historian, and I see people ask this question all the time. Plus, Weigley is my History Hero. Anyway, he says:

The short range and the inaccuracy of arquebus and musket assured the indecisiveness of exchanges of infantry fire and condemned Spanish tercios and Dutch battalions alike to a mainly defensive value in most circumstances against any reasonably well-trained and well-armed foe. In range,accuracy, and penetrating power, early hand-carried firearms represented a drastic step backward from the longbow or the crossbow of the Middle Ages. The European continent's most renowned infantry of the Middle Ages, the Swiss pikemen, had the good fortune never to confront a strong force of English longbowmen in battle. If they had, the English archers would have mowed them down. But against the first firearms, the Swiss merely dropped to the ground while the Bullets passed over their heads, then resumed the advance while the enemy reloaded. The regression in infantry missile-firing was tolerated largely because a man could become acceptably adept in handling an arquebus or musket much more quickly than he could learn to handle a longbow or crossbow properly; skill in archery usually required constant practice from early boyhood, and the decline of the English longbowmen was as much a social as a military phenomenon, involving the decline of England's independent agricultural yeomanry in the face of the first enclosure movement. Nonetheless, the superiority of the crossbow t early firearms has been estimated at forty to one[Lolwat?], and because the longbow had a considerably more rapid rate of fire than the crossbow, it superiority would have been greater yet.

Emphasis and [Color Commentary] mine.

The real issue at hand was trainability and ease of use between bows (hardest), crossbows (hard), and gunpowder weapons (ranging from hard to perhaps useable). Obviously, the bow was a niche skill that not every person could master to the level required to for competency in combat. The skill was difficult, physically demanding, and it required a level of practice and perfection that was hard to achieve in civilian life. Those skills developed (and were perhaps cultivated) among a subset of the Medieval population which began to fade as at the dawn of the Renaissance and Early Modern periods. Instead, poorer, unskilled peasants were pressed into military service. They had few skills, lacked training, and certainly they would not have had experience with a military quality bow or crossbow. Even a crossbow, tightened with a windlass or a pulley, still required strength and dexterity to tighten and shoot accurately. An Arquebus, on the other hand, required little strength, and it wasnt going to hit anything anyway. You didnt have to teach a peasant how to shoot accurately, because the gun simply wasnt capable of that. With a Crossbow, or even a longbow, their accuracy and rate of fire are wasted if the shooter couldnt actually shoot accurately. And, as Weigley pointed out, the people who developed the skills required to use these weapons properly had been marginalized by social and political trends. And why preserve and nurture those men with niche military skills? You could hire a peasant, give him a gun, and shove him off for a fraction of the cost, both monetarily and politically.

I also think youre on to something with the logistic cost a gun vs. a bow or crossbow. Fletching was another specialized skill that was A) expensive and B) difficult. And it also played a huge factor in the accuracy of the weapon in question. So, in terms of supplying an army in the field, bows and crossbows had an extra cost that was lessened by the introduction of gunpowder weaponry. But then again gunpowder itself was still a pretty new thing, and there was a lot of innovation required to make gunpowder cheap, effective, and transportable.

Sources

Russell Weigley, The Age of Battles: The Quest for Decisive Warfare from Breitenfeld to Waterloo (Bloomington: Indiana University Press, 1991), 14-15.

Well this depends on what society you're talking about. My area of expertise (if you can call it that) is Sengoku Japan. In Sengoku Era Japan, war was essentially a constant. Guns first arrived in Japan in 1543, via a Portuguese trading ship. No one in Japan really used cross bows, and actual bows require a great deal of training to use, so warlords quickly latched onto guns. They were simple to use and could be given to soldiers to fire in mass, which went along nicely with the fact that Japan had begun using formation fighting around this period. Japan had also strayed away from professional warriors and began using soldiers called Ashigaru. Ashigaru could be peasants, mercenaries, or any number of people you wouldn't typically associate with warring.

In fact, there are some historians that estimate that there were as many guns in Japan as there were in the whole of Europe during this time period.