A gunner, when compared to a bowman is really more a question of cost. Until enclosed cartridges, rifling and repeating firearms came into being, the Bow was clearly the superior weapon. The only problem was, that the training required to make a gunner was far shorter and far cheaper than the training required for a Bowman.
To prove the superiority of the bow over most muskets, look at the rate of fire, accuracy and range of a bow versus early firearms. A company of longbowman pulled from the fields of crecy or Agincourt would DECIMATE an equivilant number of british guards pulled from the fields of waterloo. It's no contest. During the American revolution, Ben Franklin tried but failed to convince the continental army to adopt the longbow as opposed to the musket to fight the brits. The British army during the same war were toying with the concept of an "experimental regiment of bowman". The reason why both projects were abandoned was because of the difficulty of training and time.
The mastery of the bow is a lifelong pursuit. The mastery of a handgonne can be done in months. That is to say normal gunners. The masters in charge of powder consistancy and quality are different, as that is much harder. But your average handgunner is by and large less expensive than his equivilant armed with a bow.