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 Breakfast Martini has gin, triple sec and orange marmalade. In parts, a Bourbon Renewal is 4 whiskey, 2 lemon, 1 cassis and 1 syrup; a Breakfast Martini is 3½ gin, 1 triple sec, 1 lemon and ¾ orange marmalade. By volume, the Bourbon Renewal is 50% spirit, 25% citrus, 13% syrup and sugar and 12% liqueur; the Breakfast Martini is 54% spirit, 16% liqueur, 16% citrus and 14% syrup and sugar. That makes them close cousins on the ratio map.
The Breakfast Martini is the stronger drink in the glass (17.6% ABV against 15.0%), but the Bourbon Renewal is the bigger drink, so it holds more alcohol (1.68 standard drinks against 1.47). The Bourbon Renewal has more calories (about 231 against 182) and more sugar (16.0 g against 8.9 g). Both are shaken; the Bourbon Renewal is served in a rocks glass, the Breakfast Martini in a martini glass.