The difference, by the numbers
Both have Angostura bitters. Only the Japanese Cocktail has Cognac, orgeat and citrus peel; only the Old Fashioned has bourbon, sugar and water. In parts, a Japanese Cocktail is 4 brandy and 1 orgeat; an Old Fashioned is 7 whiskey and 1 sugar. By volume, the Japanese Cocktail is 80% spirit and 20% syrup and sugar; the Old Fashioned is 84% spirit, 12% syrup and sugar and 4% mixer and juice. That makes them close cousins on the ratio map.
The Old Fashioned is the stronger drink in the glass (32.9% ABV against 22.0%), but the Japanese Cocktail is the bigger drink, so it holds more alcohol (1.42 standard drinks against 1.20). The Japanese Cocktail has more calories (about 178 against 133) and more sugar (9.0 g against 4.0 g). The Japanese Cocktail is stirred and served in a coupe; the Old Fashioned is built and served in a rocks glass.