The difference
Gin, lemon and syrup, shaken and served up. The Gin Basil Smash shakes in fresh basil; the Earl Grey MarTEAni infuses the gin with tea and adds egg white for foam. The Smash has more gin and is the stronger drink; the MarTEAni is sweeter.
By the numbers
The Earl Grey MarTEAni uses egg white where the Gin Basil Smash uses basil; both have lemon juice, simple syrup and gin. In parts, an Earl Grey MarTEAni is 1 lemon, 1¼ syrup and 2 gin; a Gin Basil Smash is 2¾ gin, 1 lemon and 1 syrup. By volume, the Earl Grey MarTEAni is 46% spirit, 31% syrup and sugar and 23% citrus; the Gin Basil Smash is 57% spirit, 22% syrup and sugar and 21% citrus.
The Gin Basil Smash is the stronger drink in the glass (14.1% ABV against 9.6%) and holds more alcohol (1.35 standard drinks against 1.01). Calories are close (about 192 and 191), but the Earl Grey MarTEAni has more sugar (19.2 g against 14.4 g). Both are shaken; the Earl Grey MarTEAni is served in a martini glass, the Gin Basil Smash in a coupe.