Hey Rob, good question. If I may offer my limited knowledge, and give you a basic answer, then others can give more details later
From what I've learned, there was not much difference in weaponry of the two armies. Both sides made use of the 1855 Springfield rifle, a .58 cal. musket, and the 1842 Springfield smoothbore musket, .69 cal. But when the war broke out, American arsenals didn't have enough guns to be distributed to the large number of troop demand. So the 1853 Enfield, .577 cal., imported from Britain became widely used on both sides North and South to fill out the ranks.
The North made an improvement on the 1855 rifle, which was the 1861 Springfield, .58 cal., which became the number one gun to be used by the Union; even Southern troops used it too. Second to this gun was the Enfield. For the North, the 1855 rifle and 1842 smoothbore, along with the Austrian Lorenz rifle, .54 cal., and other contract 1861 Springfields, and later editions of the 1861 Springfield, all these were not in wide use. As I said, somebody else may chime in exacts.
For the South, of course the '53 Enfield was widely used, as well as the Richmond rifle, .58 cal., which paralleled its Northern counterpart, the Springfield. But the main difference was that the Confederates still made use of the 1842 Springfield smoothbore. There are some accounts of some poorly supplied Confederate troops at the battle of Shiloh who used shotguns, and flintlock muskets.