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 Mai Tai has Jamaican rum, aged rum, orange curaçao, orgeat and lime juice. In parts, a Bourbon Renewal is 4 whiskey, 2 lemon, 1 cassis and 1 syrup; a Mai Tai is 2 rum, 2 rum, 1 Curaçao, 1 orgeat, 2 lime and ½ syrup. By volume, the Bourbon Renewal is 50% spirit, 25% citrus, 13% syrup and sugar and 12% liqueur; the Mai Tai is 47% spirit, 23% citrus, 18% syrup and sugar and 12% liqueur. 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%), but being smaller it holds about the same alcohol (1.68 and 1.69 standard drinks). The Mai Tai has more calories (about 242 against 231) and more sugar (17.2 g against 16.0 g). Both are shaken and served in a rocks glass.