The difference, by the numbers
Both have sweet vermouth. Only the Brooklyn has rye whiskey, Amer Picon and Maraschino liqueur; only the Hanky Panky has gin and Fernet-Branca. In parts, a Brooklyn is 1 whiskey and 1 vermouth; a Hanky Panky is 1 gin and 1 vermouth. By volume, the Brooklyn is 50% fortified wine and bitter and 49% spirit; the Hanky Panky is 54% fortified wine and bitter and 46% spirit. That makes them close cousins on the ratio map.
The Brooklyn is the stronger drink in the glass (20.8% ABV against 20.0%), but being smaller it holds about the same alcohol (1.57 and 1.59 standard drinks). Calories and sugar are close: about 182 and 184 calories, 7.1 g and 7.1 g of sugar. Both are stirred and served in a coupe.