The difference, by the numbers
They have no ingredient in common: the Eggnog has whole egg, sugar, Cognac, aged rum, milk and cream, the Hot Toddy bourbon, honey and water. In parts, an Eggnog is 2 sugar, 1 brandy, 1 rum, 3 milk and 2 cream; a Hot Toddy is 2 whiskey, 1 honey and 5½ water. By volume, the Eggnog is 55% mixer and juice, 23% syrup and sugar and 22% spirit; the Hot Toddy is 64% mixer and juice, 24% spirit and 12% syrup and sugar. Different bottles, similar balance: that makes them close cousins on the ratio map.
The Hot Toddy is the stronger drink in the glass (10.8% ABV against 8.5%), but the Eggnog is the bigger drink, so it holds more alcohol (1.35 standard drinks against 1.14). The Eggnog has more calories (about 612 against 209) and more sugar (44.0 g against 26.2 g). Both are built; the Eggnog is served in a coupe, the Hot Toddy in an Irish coffee glass.