The difference, by the numbers
Both have simple syrup. Only the Bourbon Renewal has bourbon, lemon juice, crème de cassis and Angostura bitters; only the Illegal has mezcal, overproof white rum, falernum, Maraschino liqueur and lime juice. In parts, a Bourbon Renewal is 4 whiskey, 2 lemon, 1 cassis and 1 syrup; an Illegal is 2 mezcal, 1 rum, 1 falernum, ¼ Maraschino, 1½ lime and 1 syrup. By volume, the Bourbon Renewal is 50% spirit, 25% citrus, 13% syrup and sugar and 12% liqueur; the Illegal is 44% spirit, 22% citrus, 19% liqueur and 15% syrup and sugar. That makes them close cousins on the ratio map.
The Bourbon Renewal is the stronger drink in the glass (15.0% ABV against 14.5%) and holds more alcohol (1.68 standard drinks against 1.43). The Bourbon Renewal has more calories (about 231 against 204) and more sugar (16.0 g against 14.7 g). Both are shaken; the Bourbon Renewal is served in a rocks glass, the Illegal in a coupe.