The difference, by the numbers
Both have lime juice. Only the Chartreuse Swizzle has Green Chartreuse, pineapple juice and falernum; only the Rum Runner has blackberry brandy, banana liqueur, grenadine and 151-proof rum. In parts, a Chartreuse Swizzle is 3 Chartreuse, 2 pineapple, 1½ lime and 1 falernum; a Rum Runner is 1¾ lime, 1 blackberry brandy, 1 banana liqueur, ¾ grenadine and ¾ rum. By volume, the Chartreuse Swizzle is 53% liqueur, 27% mixer and juice and 20% citrus; the Rum Runner is 38% liqueur, 32% citrus, 16% spirit and 14% syrup and sugar. That makes them close cousins on the ratio map.
The Chartreuse Swizzle is the stronger drink in the glass (15.6% ABV against 14.7%), but the Rum Runner is the bigger drink, so it holds more alcohol (1.84 standard drinks against 1.49). The Rum Runner has more calories (about 321 against 229) and more sugar (27.8 g against 18.6 g). The Chartreuse Swizzle is swizzled and served in a highball; the Rum Runner is blended and served in a hurricane glass.