The difference, by the numbers
Both have vodka. Only the Bloody Mary has tomato juice, lemon juice, Worcestershire sauce and Tabasco, celery salt and pepper; only the Sea Breeze has cranberry juice and grapefruit juice. In parts, a Bloody Mary is 3 vodka, 6 tomato and 1 lemon; a Sea Breeze is 1 vodka, 3 cranberry and ¾ grapefruit. By volume, the Bloody Mary is 60% mixer and juice, 30% spirit and 10% citrus; the Sea Breeze is 63% mixer and juice, 21% spirit and 16% citrus. That makes them close cousins on the ratio map.
The Bloody Mary is the stronger drink in the glass (9.1% ABV against 7.7%) and holds more alcohol (1.01 standard drinks against 0.90). The Sea Breeze has more calories (about 170 against 118) and more sugar (18.1 g against 2.7 g). Both are served in a highball; the Bloody Mary is stirred, the Sea Breeze built.