The difference, by the numbers
Both have simple syrup and gin. Only the Gin-Gin Mule has lime juice, mint and ginger beer; only the Long Island Iced Tea has vodka, tequila, white rum, triple sec, lemon juice and cola. In parts, a Gin-Gin Mule is ¾ lime, 1 syrup, 1¾ gin and 1 ginger beer; a Long Island Iced Tea is ½ vodka, ½ tequila, ½ rum, ½ gin, ½ triple sec, ¾ lemon, 1 syrup and 1¼ cola. By volume, the Gin-Gin Mule is 39% spirit, 22% syrup and sugar, 22% mixer and juice and 17% citrus; the Long Island Iced Tea is 35% spirit, 23% mixer and juice, 18% syrup and sugar, 15% citrus and 9% liqueur. That makes them close cousins on the ratio map.
The Long Island Iced Tea is the stronger drink in the glass (16.4% ABV against 10.4%) and holds more alcohol (1.69 standard drinks against 1.18). The Long Island Iced Tea has more calories (about 275 against 205) and more sugar (27.0 g against 21.9 g). Both are served in a highball; the Gin-Gin Mule is shaken, the Long Island Iced Tea built.