The difference
Both are sours shaken with lemon, syrup and egg white: the Pisco Sour with pisco, the Earl Grey MarTEAni with gin infused with Earl Grey tea. The Pisco Sour has more spirit and more alcohol; the MarTEAni more syrup.
By the numbers
The Earl Grey MarTEAni uses gin where the Pisco Sour uses pisco; both have lemon juice, simple syrup and egg white. In parts, an Earl Grey MarTEAni is 1 lemon, 1¼ syrup and 2 gin; a Pisco Sour is 3 pisco, 1½ lemon and 1 syrup. By volume, the Earl Grey MarTEAni is 46% spirit, 31% syrup and sugar and 23% citrus; the Pisco Sour is 55% spirit, 27% citrus and 18% syrup and sugar.
The Pisco Sour is the stronger drink in the glass (11.2% ABV against 9.6%) and holds more alcohol (1.35 standard drinks against 1.01). The Pisco Sour has more calories (about 203 against 192), but the Earl Grey MarTEAni has more sugar (19.2 g against 13.3 g). Both are shaken; the Earl Grey MarTEAni is served in a martini glass, the Pisco Sour in a coupe.