The difference
Bourbon, lemon and sugar, shaken, in both. The Bourbon Renewal adds crème de cassis and a dash of Angostura and is served over ice, which makes it fruitier and darker; the Whiskey Sour is plainer, with an optional egg white for foam.
By the numbers
Both have bourbon, lemon juice and simple syrup. Only the Bourbon Renewal has crème de cassis and Angostura bitters. In parts, a Bourbon Renewal is 4 whiskey, 2 lemon, 1 cassis and 1 syrup; a Whiskey Sour is 2¼ whiskey, 1¼ lemon and 1 syrup. By volume, the Bourbon Renewal is 50% spirit, 25% citrus, 13% syrup and sugar and 12% liqueur; the Whiskey Sour is 50% spirit, 28% citrus and 22% syrup and sugar.
The Bourbon Renewal is the stronger drink in the glass (15.0% ABV against 13.4%) and holds more alcohol (1.68 standard drinks against 1.14). The Bourbon Renewal has more calories (about 231 against 168) and more sugar (16.0 g against 13.0 g). Both are shaken; the Bourbon Renewal is served in a rocks glass, the Whiskey Sour in a coupe.