The difference, by the numbers
Both have lemon juice. Only the Earl Grey MarTEAni has simple syrup, gin and egg white; only the Trinidad Sour has Angostura bitters, orgeat and rye whiskey. In parts, an Earl Grey MarTEAni is 1 lemon, 1¼ syrup and 2 gin; a Trinidad Sour is 2 orgeat, 1½ lemon and 1 whiskey. By volume, the Earl Grey MarTEAni is 46% spirit, 31% syrup and sugar and 23% citrus; the Trinidad Sour is 45% syrup and sugar, 33% citrus and 22% spirit. That makes them close cousins on the ratio map.
The Trinidad Sour is the stronger drink in the glass (14.6% ABV against 9.6%) and holds more alcohol (1.51 standard drinks against 1.01). The Trinidad Sour has more calories (about 231 against 192), with about the same sugar (19.2 g and 18.6 g). Both are shaken; the Earl Grey MarTEAni is served in a martini glass, the Trinidad Sour in a coupe.