The difference
Gin, lemon and syrup in both. The John Collins is built tall over ice with soda; the Earl Grey MarTEAni infuses the gin with tea, shakes it with egg white and is served up, and is the sweeter drink.
By the numbers
The Earl Grey MarTEAni uses egg white where the John 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 John Collins is 3 gin, 2 lemon, 1 syrup and 4 soda. By volume, the Earl Grey MarTEAni is 46% spirit, 31% syrup and sugar and 23% citrus; the John Collins is 40% mixer and juice, 30% spirit, 20% citrus and 10% syrup and sugar.
The John Collins is the stronger drink in the glass (10.9% ABV against 9.6%), but being smaller it holds about the same alcohol (1.01 standard drinks each). The Earl Grey MarTEAni has more calories (about 192 against 142) and more sugar (19.2 g against 10.0 g). The Earl Grey MarTEAni is shaken and served in a martini glass; the John Collins is built and served in a highball.