The difference, by the numbers
Both have lime juice. Only the Lemon, Lime and Bitters has lemon-lime soda and alcohol-free aromatic bitters; only the Michelada has Worcestershire sauce, hot sauce, beer and salt. In parts, a Lemon, Lime and Bitters is 1 lime and 10 lemon-lime soda; a Michelada is 1 lime and 6 beer. By volume, the Lemon, Lime and Bitters is 91% mixer and juice and 9% citrus; the Michelada is 86% mixer and juice and 14% citrus. That makes them close cousins on the ratio map.
The Michelada is the stronger drink in the glass (3.8% ABV against 0.0%) and holds more alcohol (0.99 standard drinks against 0.00). The Michelada has more calories (about 171 against 68), but the Lemon, Lime and Bitters has more sugar (16.5 g against 1.1 g). Both are built and served in a highball.