For the life of me, I can't seem to discern a difference. In the 18th and early 19th century, a frigate was a much larger vessel than the punier sloop, but by the eve of the Civil War, they seem very similar. One US Navy ship, the Cumberland, was called a frigate at commissioning and renamed a sloop after a refit, but she was a large ship and carried twenty-four heavy guns, more than some frigates.
In general, it's where the ship carried its guns. A sloop-of-war, which is the techinical term for it around the time of 1860s, would carry all its armament on a single gun deck. You can see that in this image of the USS Constellation docked in Baltimore harbor. Constellation's primary armament was 8" shell-firing guns, so even though it carried only 16 of them (along with some long guns and chasers), it would have been substantially more than a match for its namesake or other contemporary frigates (e.g. the Constitution, etc.). It displaces around 1,400 long tons.
By contrast, the frigates of the Civil War were quite a bit larger (the Franklin class were ~3,400 ton ships) with armament on multiple decks, mixed steam and sail propulsion, and larger guns overall -- most carried at least one large rifle and several howitzers, as well as traditional smoothbore guns.
