The difference, by the numbers
They have no ingredient in common: the Chartreuse Swizzle has Green Chartreuse, pineapple juice, lime juice and falernum, the Grasshopper crème de cacao, crème de menthe and cream. In parts, a Chartreuse Swizzle is 3 Chartreuse, 2 pineapple, 1½ lime and 1 falernum; a Grasshopper is 1 cacao, 1 menthe and 1 cream. By volume, the Chartreuse Swizzle is 53% liqueur, 27% mixer and juice and 20% 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 Chartreuse Swizzle is the stronger drink in the glass (15.6% ABV against 10.3%) and holds more alcohol (1.49 standard drinks against 0.52). The Chartreuse Swizzle has more calories (about 229 against 179) and more sugar (18.6 g against 15.6 g). The Chartreuse Swizzle is swizzled and served in a highball; the Grasshopper is shaken and served in a coupe.