The difference, by the numbers
They have no ingredient in common: the Hemingway Special has white rum, grapefruit juice, Maraschino liqueur and lime juice, the Jasmine gin, triple sec, Campari and lemon juice. In parts, a Hemingway Special is 4 rum, 2¾ grapefruit, 1 Maraschino and 1 lime; a Jasmine is 2 gin, ¼ triple sec, ¼ Campari and 1 lemon. By volume, the Hemingway Special is 46% spirit, 42% citrus and 12% liqueur; the Jasmine is 55% spirit, 27% citrus, 9% fortified wine and bitter and 9% liqueur. Different bottles, similar balance: that makes them close cousins on the ratio map.
The Jasmine is the stronger drink in the glass (16.3% ABV against 13.8%), but the Hemingway Special is the bigger drink, so it holds more alcohol (1.62 standard drinks against 1.28). The Hemingway Special has more calories (about 194 against 146) and more sugar (7.8 g against 4.3 g). Both are shaken; the Hemingway Special is served in a coupe, the Jasmine in a martini glass.