The difference, by the numbers
Both have sugar. Only the Old Fashioned has bourbon, Angostura bitters and water; only the Sazerac has Cognac, absinthe and Peychaud’s bitters. In parts, an Old Fashioned is 7 whiskey and 1 sugar; a Sazerac is 8 brandy and 1 sugar. By volume, the Old Fashioned is 84% spirit, 12% syrup and sugar and 4% mixer and juice; the Sazerac is 87% spirit and 11% syrup and sugar. That makes them close cousins on the ratio map.
The Old Fashioned is the stronger drink in the glass (32.9% ABV against 25.2%), but being smaller it holds about the same alcohol (1.20 and 1.19 standard drinks). Calories and sugar are close: about 133 and 132 calories, 4.0 g and 4.0 g of sugar. Both are served in a rocks glass; the Old Fashioned is built, the Sazerac stirred.