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 Gin Rickey lime juice, gin and soda water. In parts, a Bloody Mary is 3 vodka, 6 tomato and 1 lemon; a Gin Rickey is 1 lime, 3 gin and 4 soda. By volume, the Bloody Mary is 60% mixer and juice, 30% spirit and 10% citrus; the Gin Rickey is 50% mixer and juice, 38% spirit and 12% citrus. Different bottles, similar balance: that makes them close cousins on the ratio map.
The Gin Rickey is the stronger drink in the glass (13.3% ABV against 9.1%), but being smaller it holds about the same alcohol (1.01 standard drinks each). The Bloody Mary has more calories (about 118 against 103) and more sugar (2.7 g against 0.3 g). Both are served in a highball; the Bloody Mary is stirred, the Gin Rickey built.