This is a copy-paste of a response I gave a while back to someone asking whether bows and arrows would have been effective during the Revolutionary War, explaining that when smoothbore matchlocks fully replaced bows in the 16th century many people did consider the gun's to have far better range and accuracy than bows:

French General Baron de Marbot wrote about his encounters with Tartar and Baskir horse archers fighting for the Russians during the Napoleonic wars. Overall they were found to be very unimpressive, even though the French wore almost no armor. He also gives an example of why the ability of archers to fire over the heads of troops in front of them isn't as much of an advantage as you would think.

I have never understood with what aim the Russian government brought from so far and at such great expense these masses of irregular cavalry, who having neither sabres nor lances nor any kind of firearm, were unable to stand up against trained soldiers, and served only to strip the countryside and starve the regular forces, which alone were capable of resisting a European enemy. Our soldiers were not in the least alarmed at the sight of these semi-barbarous Asiatics, whom they nicknamed cupids, because of their bows and arrows.

Nevertheless, these newcomers, who did not yet know the French, had been so indoctrinated by their leaders, almost as ignorant as themselves, that they expected to see us take flight at their approach; and so they could not wait to attack us. From the very day of their arrival in sight of our troops they launched themselves in swarms against them, but having been everywhere repulsed by gunfire, the Baskirs left a great number of dead on the ground.

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.

The supposed superiority of bows and arrows over even smoothbore firearms in terms of range and accuracy is likely greatly overstated. During the late 16th century the abandonment of the longbow by the English army was a very controversial and a great deal was written by experienced military men both for and against the switch. Some, like Sir John Smythe did indeed feel that England's ancestral weapon was vastly superior to firearms and that muskets were completely useless beyond 20-30 yards. But many others, such as Roger Williams, claimed that a musket with good powder was deadly to both horse and man at up to 600 yards. According to him, bowmen were "the worst shot used these days" and in the field 1500 bowmen would be less effective than 500 musketmen. Humfrey Barwick, an English soldier who served as a mercenary on the continent wrote a lengthy rebuttal to many of Smythe's arguments and claimed that he personally could hit a single man with his arquebus at 120 yards while doubting that the best archers in england at the time could do the same. Obviously not every musketman could achieve this, especially in the heat of battle, but bows would be even more difficult to aim. As Barnabe Rich put it, a bow may miss wide, short, or long while a firearm may only miss wide. What seems to be a common thread is even bow proponents noting that the number of archers skilled enough to reliably outshoot a skilled man firearm seemed to be fairly small, and dropping rapidly. Barwick took this one step further and noted that in his experience archers had a tendency to rapidly grow tired in combat and stop bringing their arrows to full draw, or that poor nutrition and sleeping in the cold might leave them too weak to shoot their bows at all.

A couple other arguments made in favor of firearms in the late 16th century:

-A flatter trajectory meant that a standing man presented a larger target, thus a bullet was more likely to hit than an arrow falling down at an angle.

-The weakest guns tended to have far more force than even the strongest bows. Unless you are using an extremely high draw-weight bow it might not be enough to break bones or pierce a man's skull or ribcage. Only the thickest armors could stop most firearms.

-Guns could be used more easily from cover than bows could and could be kept in a ready to fire position.

-At shorter ranges muskets could be loaded with multiple projectiles. This is especially relevant to the revolutionary war when double-shotting muskets or loading them with "buck and ball" shot was extremely common. Even in terms of volume of fire a formation of archers would likely still be at a disadvantage against a line of musketmen firing 4-5 shots per minute and 2-5 projectiles per shot.

Sources

The Memoirs of General the Baron de Marbot http://www.gutenberg.org/cache/epub/2401/pg2401-images.html

Certain Discourse Written by Sir John Smythe http://quod.lib.umich.edu/e/eebo/A12567.0001.001?view=toc

Roger Williams. A Briefe Discourse of Warre http://quod.lib.umich.edu/e/eebo/A15466.0001.001?view=toc

Humfrey Barwick. A breefe discourse, concerning the force and effect of all manuall weapons of fire http://quod.lib.umich.edu/e/eebo/A05277.0001.001?view=toc

Barnabe Rich. A right exelent and pleasaunt dialogue http://quod.lib.umich.edu/e/eebo/A10718.0001.001?view=toc

Please take a look at the FAQ section on early firearms here

Just a quick summary, it was far easier to train a person to use a gun than a bow (and an unsourced claim that the gun was likely easier to manufacture than the crossbow). By the late 16th century, when bows really disappeared, guns also packed more punch than bows and and most crossbows.

Just a note, while it was possible for archers to loose an arrow every few seconds, it was unlikely for them to have done so in battle as they would have quickly ran out of arrows.