The difference, by the numbers
Both have amaretto and lemon juice. Only the Alabama Slammer has Southern Comfort and sloe gin; only the Amaretto Sour has Angostura bitters and egg white. In parts, an Alabama Slammer is 2 amaretto, 2 Southern Comfort, 1 sloe gin and ½ lemon; an Amaretto Sour is 2 amaretto and 1 lemon. By volume, the Alabama Slammer is 91% liqueur and 9% citrus; the Amaretto Sour is 67% liqueur and 33% citrus. That makes them close cousins on the ratio map.
The Alabama Slammer is the stronger drink in the glass (22.1% ABV against 10.7%) and holds more alcohol (1.28 standard drinks against 0.97). Calories are close (about 188 and 182), but the Amaretto Sour has more sugar (18.9 g against 15.2 g). The Alabama Slammer is built and served in a highball; the Amaretto Sour is shaken and served in a rocks glass.