The difference
Gin, lemon and syrup in both. The Fitzgerald seasons them with Angostura and is served over ice; the Earl Grey MarTEAni uses tea-infused gin and an egg white for foam, and is served up.
By the numbers
The Earl Grey MarTEAni uses egg white where the Fitzgerald uses Angostura bitters; both have lemon juice, simple syrup and gin. In parts, an Earl Grey MarTEAni is 1 lemon, 1¼ syrup and 2 gin; a Fitzgerald is 2 gin, 1¼ syrup and 1 lemon. By volume, the Earl Grey MarTEAni is 46% spirit, 31% syrup and sugar and 23% citrus; the Fitzgerald is 46% spirit, 31% syrup and sugar and 23% citrus. That makes them close cousins on the ratio map.
The Fitzgerald is the stronger drink in the glass (12.2% ABV against 9.6%), but being smaller it holds about the same alcohol (1.01 and 1.05 standard drinks). The Earl Grey MarTEAni has more calories (about 192 against 180), with about the same sugar (19.2 g and 19.0 g). Both are shaken; the Earl Grey MarTEAni is served in a martini glass, the Fitzgerald in a rocks glass.