The difference, by the numbers
They have no ingredient in common: the Navy Grog has lime juice, grapefruit juice, soda water, dark rum, Jamaican rum, white rum and honey syrup, the New York Sour rye whiskey, simple syrup, lemon juice, egg white and red wine. In parts, a Navy Grog is ¾ lime, ¾ grapefruit, ¾ soda, 1 rum, 1 rum, 1 rum and 1 honey; a New York Sour is 4 whiskey, 1½ syrup, 2 lemon and 1 red wine. By volume, the Navy Grog is 48% spirit, 24% citrus, 16% syrup and sugar and 12% mixer and juice; the New York Sour is 47% spirit, 23% citrus, 18% syrup and sugar and 12% mixer and juice. Different bottles, similar balance: that makes them close cousins on the ratio map.
The New York Sour is the stronger drink in the glass (14.2% ABV against 12.3%), but the Navy Grog is the bigger drink, so it holds more alcohol (2.03 standard drinks against 1.64). The Navy Grog has more calories (about 279 against 225) and more sugar (20.0 g against 14.7 g). Both are shaken and served in a rocks glass.