If you're asking why they didn't use heavy warbows like the english longbow, then the short answer is that the social infrastructure and tradition needed to produce large numbers of soldiers capable of drawing 100-180 lb bows basically no longer existed. The concept of recreating corps of longbowmen like those from the Hundred Years War to fight in modern conflicts was occasionally brought up by various writers in the 17th-19th centuries, notably Ben Franklin during the Revolution, but generals at this time couldn't field such troops even if they wanted too (in fact it seems that Franklins' recommendation to fill the colonial army with companies of pikemen was taken much more seriously than his archery suggestion).

The more complex answer is that bows and arrows likely wouldn't have helped much anyways. Even when archery was still widely practiced the question of whether bows or matchlock muskets were more accurate in a direct comparison seems to have been debatable at best. Writing about his early years as in the English army in the mid 1500s, Humfrey Barwick claimed that his commander held trials of arms 2-3 times a week involving longbows, crossbows, and firearms which wound up leaving little doubt that firearms were the most accurate of the three. He also concluded that after just a few months of training with his arquebus he could hit any target near or far as well as "the best archers in England." When the matchlock musket arrived in China and Korea, local sources wrote that on the practice field they would strike the target "five times better than the bow and arrow."

A soldier's musket like those used during the revolutionary war could supposedly strike the the figure of a man at 80 yards, a feat that takes quite a lot of practice to match with a bow and arrow. A hunting musket, which used a ball that fit the barrel more tightly and a much smaller powder charge, was likely even more accurate and that's not even getting into the American long rifle, which could supposedly strike a man's head at 200 yards most of the time. In late 18th century trials it was found that volleys of musket fire could inflict casualties on a battalion-sized target 400 or even 500 yards away, which is even more so exceedingly difficult to match with longbows. In actual combat the accuracy was always much lower than on the practice range and it wasn't too uncommon to see entire musket volleys miss at 100 paces, but the fact that so much of it is attributable to human error suggests that replacing the muskets with a weapon far more complicated to aim and shoot would not have helped much.

In theory it doesn't take a lot to kill an unarmored man. Dr. Joseph Bill's "Notes on Arrow Wounds" compiled various examples of arrow wounds suffered on the american frontier in the 19th century and concluded that they were indeed quite deadly, particularly if they struck the lungs or gut, or if the arrowhead broke inside the wound and surgery was not available. However even so it seems that the sheer stopping power of a large-caliber musket ball tended to be much preferred and had a far greater psychological impact. With a full military charge an 18th century musket ball could achieve a muzzle velocity of almost 1.5 times the speed of sound, shattering bones and creating a larger wound cavity than even many modern assault rifle rounds. Arrows by contrast are unlikely to break bones and may not even pierce the skull unless shot from a powerful bow. As an example of this from the Napoleonic Wars, French general Marbot wrote about his experiences fighting Asian horse archers who had been levied by the Russian army mocking the fact that their arrows caused only "minor wounds" if they hit and claimed that he recalled seeing only one man actually killed by the weapon. Marbot himself was wounded by an arrow which penetrated "about an inch" into his thigh. He claimed that he didn't even notice it until it got in the way of his sword hand, and even then it did not affect on his ability to command.

The one real advantage archers would seem to have over firearms would seem to be rate of fire. However, even that comes with caveats. A smoothbore musket could be loaded with a wide variety of ammunition, and in the americas loading a weapon with multiple bullets has a long history, dating all the way back to Samuel de Champlain who in the early 17th century was able to kill two Iroquois chiefs and wounded a third in a single shot by loading his arquebus with four bullets. During the revolutionary war, double shotting muskets or loading them with buckshot was extremely common to the point where "buck and ball" cartridges made of one musket ball and 3-4 buckshot became standard in the continental army. In terms of total projectiles it was definitely possible for guns to achieve a higher overall rate of fire than bows and arrows (the record for a punt gun is around 500 birds with a single shot) however in terms of total aimed shots then the bow could win out as long as the archer's strength holds.

Overall I highly doubt bows would have been a very effective weapon for either side during the revolution. If a bunch of american colonists were given bows and arrows and told to go fight a guerilla war against the British, it would probably go about as well as this account of John Mason fighting Pequot archers shortly after the Mystic Massacre:

The Enemy still following us in the Rear, which was to windward, though to little purpose ; yet some of them lay in Ambush behind Rocks and Trees, often shooting at us, yet through Mercy touched not one of us : And as we came to any Swamp or Thicket, we made some Shot to clear the Passage. Some of them fell with our Shot; and probably more might, but for want of Munition

minus the "want of Munition" part

Heath. Bow versus Gun

Hughes. Firepower: Weapons Effectiveness on the Battlefield 1630-1850

Lewis. Small arms and ammunition in the United States Service

Hall. Weapons and Warfare in Renaissance Europe

Andrade. The Gunpowder Age

Phillips. "Longbow and Hackbutt: Weapons Technology and Technology Transfer in Early Modern England"

Mason, John. A Brief History of the Pequot War