The difference, by the numbers
Both have lemon juice. Only the Bourbon Renewal has bourbon, crème de cassis, simple syrup and Angostura bitters; only the Jasmine has gin, triple sec and Campari. In parts, a Bourbon Renewal is 4 whiskey, 2 lemon, 1 cassis and 1 syrup; a Jasmine is 2 gin, ¼ triple sec, ¼ Campari and 1 lemon. By volume, the Bourbon Renewal is 50% spirit, 25% citrus, 13% syrup and sugar and 12% liqueur; the Jasmine is 55% spirit, 27% citrus, 9% fortified wine and bitter and 9% liqueur. That makes them close cousins on the ratio map.
The Jasmine is the stronger drink in the glass (16.3% ABV against 15.0%), but the Bourbon Renewal is the bigger drink, so it holds more alcohol (1.68 standard drinks against 1.28). The Bourbon Renewal has more calories (about 231 against 146) and more sugar (16.0 g against 4.3 g). Both are shaken; the Bourbon Renewal is served in a rocks glass, the Jasmine in a martini glass.