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 Paloma tequila, lime juice, salt and grapefruit soda. In parts, a Bloody Mary is 3 vodka, 6 tomato and 1 lemon; a Paloma is 1 tequila and 2 grapefruit soda. By volume, the Bloody Mary is 60% mixer and juice, 30% spirit and 10% citrus; the Paloma is 65% mixer and juice, 32% spirit and 3% citrus. Different bottles, similar balance: that makes them close cousins on the ratio map.
The Paloma is the stronger drink in the glass (11.7% ABV against 9.1%) and holds more alcohol (1.13 standard drinks against 1.01). The Paloma has more calories (about 154 against 118) and more sugar (10.8 g against 2.7 g). Both are served in a highball; the Bloody Mary is stirred, the Paloma built.