The difference, by the numbers
Both have lemon juice and simple syrup. Only the Bourbon Renewal has bourbon, crème de cassis and Angostura bitters; only the Bramble has gin and crème de mûre. In parts, a Bourbon Renewal is 4 whiskey, 2 lemon, 1 cassis and 1 syrup; a Bramble is 4 gin, 2 lemon, 1 syrup and 1¼ mûre. By volume, the Bourbon Renewal is 50% spirit, 25% citrus, 13% syrup and sugar and 12% liqueur; the Bramble is 49% spirit, 24% citrus, 15% liqueur and 12% 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 14.3%) and holds more alcohol (1.68 standard drinks against 1.26). The Bourbon Renewal has more calories (about 231 against 177) and more sugar (16.0 g against 12.8 g). Both are shaken and served in a rocks glass.