The difference
Whiskey, sugar and water in both. The Old Fashioned seasons them with Angostura and is built over ice; the Whiskey Smash presses in mint over shaved ice instead, like a small Mint Julep.
By the numbers
The Old Fashioned uses Angostura bitters where the Whiskey Smash uses mint; both have bourbon, sugar and water. In parts, an Old Fashioned is 7 whiskey and 1 sugar; a Whiskey Smash is 1 sugar, 1½ water and 6 whiskey. By volume, the Old Fashioned is 84% spirit, 12% syrup and sugar and 4% mixer and juice; the Whiskey Smash is 71% spirit, 18% mixer and juice and 11% syrup and sugar.
The Old Fashioned is the stronger drink in the glass (32.9% ABV against 25.4%), but the Whiskey Smash is the bigger drink, so it holds more alcohol (1.52 standard drinks against 1.20). The Whiskey Smash has more calories (about 172 against 133) and more sugar (6.0 g against 4.0 g). Both are built and served in a rocks glass.