The difference, by the numbers
Both have white rum. Only the Fog Cutter has Cognac, gin, orange juice, lemon juice, orgeat and dry sherry; only the Navy Grog has lime juice, grapefruit juice, soda water, dark rum, Jamaican rum and honey syrup. In parts, a Fog Cutter is 2 rum, 1 brandy, ½ gin, 1 orange, 2 lemon, ½ orgeat and ½ dry sherry; a Navy Grog is ¾ lime, ¾ grapefruit, ¾ soda, 1 rum, 1 rum, 1 rum and 1 honey. By volume, the Fog Cutter is 47% spirit, 26% citrus, 13% mixer and juice, 7% fortified wine and bitter and 7% syrup and sugar; the Navy Grog is 48% spirit, 24% citrus, 16% syrup and sugar and 12% mixer and juice. That makes them close cousins on the ratio map.
The Fog Cutter is the stronger drink in the glass (13.0% ABV against 12.3%) and holds more alcohol (2.52 standard drinks against 2.03). The Fog Cutter has more calories (about 314 against 279), but the Navy Grog has more sugar (20.0 g against 13.3 g). Both are shaken; the Fog Cutter is served in a highball, the Navy Grog in a rocks glass.