By the numbers
Both have gin, triple sec and lemon juice. Only the Breakfast Martini has orange marmalade. In parts, a Breakfast Martini is 3½ gin, 1 triple sec, 1 lemon and ¾ orange marmalade; a White Lady is 2 gin, 1½ triple sec and 1 lemon. By volume, the Breakfast Martini is 54% spirit, 16% liqueur, 16% citrus and 14% syrup and sugar; the White Lady is 45% spirit, 33% liqueur and 22% citrus.
They are about equally strong in the glass (17.6% and 17.8% ABV), but the White Lady is the bigger drink, so it holds more alcohol (1.58 standard drinks against 1.47). Calories and sugar are close: about 182 and 189 calories, 8.9 g and 8.0 g of sugar. Both are shaken; the Breakfast Martini is served in a martini glass, the White Lady in a coupe.