The difference, by the numbers
Both have crème de cacao. Only the Bushwacker has dark rum, coffee liqueur, cream of coconut and milk; only the Grasshopper has crème de menthe and cream. In parts, a Bushwacker is 1 rum, 1 coffee liqueur, 1 cacao, 2 coconut and 2 milk; a Grasshopper is 1 cacao, 1 menthe and 1 cream. By volume, the Bushwacker is 57% mixer and juice, 29% liqueur and 14% spirit; the Grasshopper is 67% liqueur and 33% mixer and juice. That makes them close cousins on the ratio map.
The Grasshopper is the stronger drink in the glass (10.3% ABV against 7.3%), but the Bushwacker is the bigger drink, so it holds more alcohol (1.39 standard drinks against 0.52). The Bushwacker has more calories (about 526 against 179) and more sugar (64.3 g against 15.6 g). The Bushwacker is blended and served in a hurricane glass; the Grasshopper is shaken and served in a coupe.