The difference, by the numbers
Both have sugar and cream. Only the Eggnog has whole egg, Cognac, aged rum and milk; only the Irish Coffee has Irish whiskey and coffee. In parts, an Eggnog is 2 sugar, 1 brandy, 1 rum, 3 milk and 2 cream; an Irish Coffee is 1 whiskey, 2½ coffee and 1 cream. By volume, the Eggnog is 55% mixer and juice, 23% syrup and sugar and 22% spirit; the Irish Coffee is 75% mixer and juice, 22% spirit and 3% syrup and sugar. That makes them close cousins on the ratio map.
The Irish Coffee is the stronger drink in the glass (9.0% ABV against 8.5%), but the Eggnog is the bigger drink, so it holds more alcohol (1.35 standard drinks against 1.13). The Eggnog has more calories (about 612 against 302) and more sugar (44.0 g against 5.4 g). Both are built; the Eggnog is served in a coupe, the Irish Coffee in an Irish coffee glass.