For starts, just to confirm if you don't know, the majority of muskets in the Civil War were rifles, or technically not muskets at all. The Springfield 1861, and then the Enfield 1853 were the most common firearms of the war.

As for smoothbores, the last US military smoothbore, musket-wise, was the 1842 Springfield. Your average Revolutionary War musket was generally a second model Brown Bess or French Charleville. These were .75 caliber and .69 caliber respectively, though the military would issue fairly smaller bullets, .69 caliber and .65 caliber respectively, I think. I admit I do not know the proper size of the Charleville.

The big difference between civil war muskets and revolutionary war ones was the Percussion lock system. Revolutionary War, and even Napoleonic War muskets used Flintlocks, where civil war muskets (except for rare cases) used Percussion locks. To not make this too long I'll share a couple videos that show the difference fairly well.

https://youtu.be/6LiCUUUWuFw https://youtu.be/dgUXrLvKcv0

Big thing is percussion locks generally have a faster ignition, a much more guaranteed ignition, and a "gentler" ignition, as the flintlock can blow smoke in the users face, a feature removed with the percussion lock. This all makes faster to load and a tiny bit easier to aim.

Additionally, the 2nd Model Brown Bess and military muskets in general of the time, did not have any form of sights besides using the bayonet lug, or what a bayonet would be attached to the gun with, as a makeshift front sight. The 1842 had a proper front sight, though the original didn't have a rear sight, which helps line up the gun correctly. Rear sights aren't exactly necessary but they certainly help a new shooter and older ones at longer ranges.

Generally the 1842 was loaded with Buck'n'Ball cartridges, or a musket plus three smaller balls placed in front of it. This was heavily used in the Mexican American war and was prominently used by 1842-armed civil war units. Though BucknBall existed in the rev war and was used I do not believe it was a mainstream ammunition, but do not take my word on that.

To sum up, the Civil War musket was better.