The difference, by the numbers
Both have sweet vermouth. Only the Remember the Maine has rye whiskey, cherry liqueur and absinthe; only the Rob Roy has Blended Scotch and Angostura bitters. In parts, a Remember the Maine is 4 whiskey, 1½ vermouth and 1 cherry; a Rob Roy is 2 Scotch and 1 vermouth. By volume, the Remember the Maine is 61% spirit, 24% fortified wine and bitter and 15% liqueur; the Rob Roy is 67% spirit and 33% fortified wine and bitter. That makes them close cousins on the ratio map.
The Remember the Maine is the stronger drink in the glass (23.9% ABV against 21.9%) and holds more alcohol (2.00 standard drinks against 1.68). The Remember the Maine has more calories (about 227 against 183) and more sugar (7.9 g against 4.5 g). Both are stirred and served in a coupe.