The difference, by the numbers
Both have sweet vermouth. Only the Remember the Maine has rye whiskey, cherry liqueur and absinthe; only the Tipperary has Irish whiskey, Green Chartreuse and Angostura bitters. In parts, a Remember the Maine is 4 whiskey, 1½ vermouth and 1 cherry; a Tipperary is 2 whiskey, 1 vermouth and ½ Chartreuse. By volume, the Remember the Maine is 61% spirit, 24% fortified wine and bitter and 15% liqueur; the Tipperary is 55% spirit, 28% fortified wine and bitter and 17% liqueur. That makes them close cousins on the ratio map.
They are about equally strong in the glass (both 23.9% ABV), but the Remember the Maine is the bigger drink, so it holds more alcohol (2.00 standard drinks against 1.86). The Remember the Maine has more calories (about 227 against 212), with about the same sugar (7.9 g and 7.5 g). Both are stirred; the Remember the Maine is served in a coupe, the Tipperary in a martini glass.