The difference, by the numbers
Both have Cognac. Only the Japanese Cocktail has orgeat, Angostura bitters and citrus peel; only the Sazerac has absinthe, sugar and Peychaud’s bitters. In parts, a Japanese Cocktail is 4 brandy and 1 orgeat; a Sazerac is 8 brandy and 1 sugar. By volume, the Japanese Cocktail is 80% spirit and 20% syrup and sugar; the Sazerac is 87% spirit and 11% syrup and sugar. That makes them close cousins on the ratio map.
The Sazerac is the stronger drink in the glass (25.2% ABV against 22.0%), but the Japanese Cocktail is the bigger drink, so it holds more alcohol (1.42 standard drinks against 1.19). The Japanese Cocktail has more calories (about 178 against 132) and more sugar (9.0 g against 4.0 g). Both are stirred; the Japanese Cocktail is served in a coupe, the Sazerac in a rocks glass.