Can't speak to cannons.

In 1776, muskets had been largely the same for around 100 years: flintlock, smooth-bore, muzzle-loading, firing a round ball of large caliber ( .69-.80) . The locks themselves had become more refined: the dog-lock had been superseded by a lock with a half-cock notch, but there had not been that much change. And they would stay pretty much unchanged for another 50 years or so. In the Civil War, on the other hand, though there were still mostly muzzleloading guns, the flintlock had been replaced by a percussion lock, the smoothbore often with a rifled one, the round ball with a minie bullet that didn't need a patch: so loading was faster, accurate range was farther. This was a time of change in firearms technology generally, though, and more modern guns did appear: there were some breechloading cartridge guns ( mostly in the hands of cavalry) and some pistols, like Colts and Remingtons, that could fire several shots. The US was lagging behind Europe in this, where breechloading cartridge guns had begun to appear in armies in the 1840's: immediately after the Civil War the US would adopt a breech-loading cartridge rifle as well.

However, since you ask about technology: there was an even greater difference in the manufacturing of the Revolution and the Civil War. In 1776, military contractors and armories employed large numbers of craftsmen who would standardize to some extent, with bore gauges to keep the same caliber, patterns and forging dies for making parts about the same size, but there was a great deal of hand fitting, which meant there were variations from gun to gun, and the hand labor meant production was limited. There was a recognition by the French earlier in the 18th c. that it was useful if military weapons had interchangeable parts- it made battlefield repairs much easier. This idea was brought back to the US by Thomas Jefferson after the Revolution, finally done at Harper's Ferry Armory in the Hall shop in the 1830's. By the time of the Civil War, it was standard practice. There was also less hand labor: water-powered belt-driven machinery was doing many of the tasks that craftsmen had done in 1776, like rifling, Blanchard lathes were making duplicate rifle stocks, and gauges, jigs , fixtures and machine tools were machining parts that previously had been forged, chipped and filed. So, although there was a significant jump in the design of the guns in the Civil War, from the Revolution, there was a much, much bigger jump in how they were made.