On the American frontier, the bow was more effective against muzzle loading muskets and rifles than is often credited as well (although the most feared weapon on the frontier was neither; it was the tomahawk).
For one, rate of fire is about 6 times higher with a bow. I fire muzzle loaders a lot, and am pretty quick when it comes to measuring powder and loading "out of the bag". Even so, I have trouble getting an aimed shot off in less than 40 seconds.
The early militia was expected to be able to load and fire quickly, but the vast majority of Americans weren't in the militia... Against small 1-15 man groups, and using guerrilla-type warfare, that quicker rate-of-fire was an advantage.
Another advantage was stealth. Several archers could fire once or twice each and not immediately give away their position. Dropping selected members of the opposing party, they could possibly even remove several opponents before the alarm was raised. I've had the misfortune of missing several deer while hunting with a longbow or recurve. Often the arrow flight is the loudest noise (unlike those clanky new-fangled training-wheel bows). Several times I've missed a deer from the "right", and had the deer look to the left (where the arrow hit) wondering what was up. Meanwhile, with the deer facing away I nock another arrow... I've even hit deer, and had them jump toward me, and away from the arrow that passed through it and clanked against the ground on the opposite side.
The arching flight of the arrow is another advantage, when your opponents are forted-up behind a low obstacle (fallen tree or rock). Unlike a bullet, an arrow can be "lobbed" quite effectively.
Another aspect to consider is that while animals are dying due to massive blood-loss, and can take a while to do so (traveling quite some distance), people go into shock very quickly, even from minor wounds. You can die (or at least be incapacitated) from shock, even if the wound isn't fatal...