The difference, by the numbers
Both have lime juice and honey syrup. Only the Navy Grog has grapefruit juice, soda water, dark rum, Jamaican rum and white rum; only the Three Dots and a Dash has rhum agricole, aged rum, falernum, allspice dram, orange juice and Angostura bitters. In parts, a Navy Grog is ¾ lime, ¾ grapefruit, ¾ soda, 1 rum, 1 rum, 1 rum and 1 honey; a Three Dots and a Dash is 3 rum, 1 rum, ½ falernum, ½ allspice, 1 lime, 1 orange and 1 honey. By volume, the Navy Grog is 48% spirit, 24% citrus, 16% syrup and sugar and 12% mixer and juice; the Three Dots and a Dash is 50% spirit, 13% liqueur, 13% syrup and sugar, 12% citrus and 12% mixer and juice. That makes them close cousins on the ratio map.
The Three Dots and a Dash is the stronger drink in the glass (16.3% ABV against 12.3%), but the Navy Grog is the bigger drink, so it holds more alcohol (2.03 standard drinks against 1.79). The Navy Grog has more calories (about 279 against 235) and more sugar (20.0 g against 14.5 g). The Navy Grog is shaken and served in a rocks glass; the Three Dots and a Dash is blended and served in a goblet.