The difference, by the numbers
They have no ingredient in common: the Bloody Mary has vodka, tomato juice, lemon juice, Worcestershire sauce and Tabasco, celery salt and pepper, the Cuba Libre white rum, cola and lime juice. In parts, a Bloody Mary is 3 vodka, 6 tomato and 1 lemon; a Cuba Libre is 1 rum, 2½ cola and ¼ lime. By volume, the Bloody Mary is 60% mixer and juice, 30% spirit and 10% citrus; the Cuba Libre is 67% mixer and juice, 28% spirit and 5% citrus. Different bottles, similar balance: that makes them close cousins on the ratio map.
The Cuba Libre is the stronger drink in the glass (10.1% ABV against 9.1%) and holds more alcohol (1.13 standard drinks against 1.01). The Cuba Libre has more calories (about 165 against 118) and more sugar (12.5 g against 2.7 g). Both are served in a highball; the Bloody Mary is stirred, the Cuba Libre built.