The difference, by the numbers
They have no ingredient in common: the Alabama Slammer has amaretto, Southern Comfort, sloe gin and lemon juice, the Grasshopper crème de cacao, crème de menthe and cream. In parts, an Alabama Slammer is 2 amaretto, 2 Southern Comfort, 1 sloe gin and ½ lemon; a Grasshopper is 1 cacao, 1 menthe and 1 cream. By volume, the Alabama Slammer is 91% liqueur and 9% citrus; the Grasshopper is 67% liqueur and 33% mixer and juice. Different bottles, similar balance: that makes them close cousins on the ratio map.
The Alabama Slammer is the stronger drink in the glass (22.1% ABV against 10.3%) and holds more alcohol (1.28 standard drinks against 0.52). Calories and sugar are close: about 188 and 179 calories, 15.2 g and 15.6 g of sugar. The Alabama Slammer is built and served in a highball; the Grasshopper is shaken and served in a coupe.