The difference
Gin, lemon and sugar in both. The Earl Grey MarTEAni infuses the gin with tea and shakes it with egg white and plenty of syrup for a silky, sweet drink served up; the Tom Collins is a tall, fizzy, much drier gin cooler.
By the numbers
The Earl Grey MarTEAni uses egg white where the Tom Collins uses soda water; both have lemon juice, simple syrup and gin. In parts, an Earl Grey MarTEAni is 1 lemon, 1¼ syrup and 2 gin; a Tom Collins is 1 lemon, 2 gin and 2 soda. By volume, the Earl Grey MarTEAni is 46% spirit, 31% syrup and sugar and 23% citrus; the Tom Collins is 39% mixer and juice, 39% spirit, 19% citrus and 3% syrup and sugar.
The Tom Collins is the stronger drink in the glass (11.1% ABV against 9.6%) and holds more alcohol (1.35 standard drinks against 1.01). The Earl Grey MarTEAni has more calories (about 192 against 150) and more sugar (19.2 g against 3.5 g). Both are shaken; the Earl Grey MarTEAni is served in a martini glass, the Tom Collins in a highball.