The definitions of "sloop" and "cutter" are complicated - a real morass of confusing, antiquated word usage. Anybody who claims to have a simple explanation of the subject hasn't read enough about it.
A good place to start such a discussion usually is the relevant volume in the Conway's History of the Ship series, in this case the volume titled The Line of Battle: The Sailing Warship, 1650-1840. Here are the definitions it offers:
cutter. (1) Ship's boat, usually clinker-built (qv), seaworthy and handier under sail than oars. (2) Sharp lined fast-sailing coastal craft, oritinally clinker-built in the English Channel ports; carried a single-masted fore and aft rig of large area, and popular as smuggling craft, revenue cruiser and small warship.
sloop. Originally a boat designation (see shallop), the term came to have two broad areas of meaning: (1) In the Royal Navy it developed into a small cruiser, below the six Rates and commanded by a Master and Commander; it virtually became the seventh Rate, being applied to any vessel so commanded, such as the larger brigs and even fireships and bomb vessels when assigned cruising duties. (2) As a rig it denoted a single-masted gaff rigged vessel with fixed bowsprit and jib headsails, and usually no square topsails.
The latter defiinition alludes to one source of confusion: the fact that the same word was used on the one hand to define a vessel's rig, and on the other to define its function.
I've seen many references to the idea that a cutter had a sliding bowsprit, whereas a sloop didn't. And of course both cutters and sloops frequently had square sails. (I don't think I've ever seen a picture of a British naval or revenue cutter that didn't.) I think the terms were used pretty carelessly in the eighteenth and nineteenth centuries. And that Anatomy of the Ship volume on the Alert mentions that, at one point in her career, she was officially reclassified from "cutter" to "sloop" because (drum roll, please) her commanding officer had been promoted from lieutenant to master and commander.
The United States government bastardized the word "cutter" even more brutally when, in 1794, the Congress, at the behest of Alexander Hamilton, the first Secretary of the Treasury, authorized the construction of thirteen cutters for the purspose of collecting the newly-authorized tariffs on imported goods. All thirteen of the vessels in question turned out to be two-masted schooners. But apparently, due to British practice, the words "revenue" and "cutter" had become inextricably connected. The old U.S. Revenue Cutter Service's modern successor, the U.S. Coast Guard, still officially applies the word "cutter" to all vessels longer than a hundred feet.