The difference, by the numbers
They have no ingredient in common: the Brandy Old Fashioned has sugar, Angostura bitters, orange juice, cocktail cherries, Cognac and lemon-lime soda, the Mary Pickford white rum, pineapple juice, Maraschino liqueur and grenadine. In parts, a Brandy Old Fashioned is 1 brandy and 1 lemon-lime soda; a Mary Pickford is 1 rum and 1 pineapple. By volume, the Brandy Old Fashioned is 48% mixer and juice, 47% spirit and 5% syrup and sugar; the Mary Pickford is 44% mixer and juice, 44% spirit, 7% liqueur and 5% syrup and sugar. Different bottles, similar balance: that makes them close cousins on the ratio map.
The Brandy Old Fashioned is the stronger drink in the glass (17.1% ABV against 12.7%) and holds more alcohol (1.43 standard drinks against 1.15). The Brandy Old Fashioned has more calories (about 190 against 163) and more sugar (12.4 g against 9.6 g). The Brandy Old Fashioned is built and served in a rocks glass; the Mary Pickford is shaken and served in a coupe.