Even though they are nearly 100 years apart, I've always considered them to have been fought with (seemingly) very similar weaponry. But a Centuary is a long time, so I would assume a lot had changed. What significant advancements had been made between the two wars?
Whew, where to start?
Rifled muskets (Springfield 1861 accurate to 400 yards, 18th century Brown Bess accurate to 100 yards)
Minie ball
Percussion cap instead of flintlock
Revolvers
Breech-loading rifles (Sharps)
Henry lever action repeating rifle (not wide-spread, but there)
Telescopic sight
Ironclads
Steamships
Armstrong screw (breech-loading cannon)
Logistics development
Railroads
Railroads
Railroads
Railroads
Industry being able to quickly produce new things, like going from primarily smooth-bore to rifled in a year or two.
Codified battlefield visual signaling (wig-wag, go Signal Corps!)
Lots and lots of things changed. The only thing that really didn't change is that line units were predominately muzzle-loaders. but that's like saying the BAR is the same as an M16.
edit: omg formatting.