The difference, by the numbers
Both have gin and sweet vermouth. Only the Bronx has dry vermouth, orange juice and orange bitters; only the Hanky Panky has Fernet-Branca. In parts, a Bronx is 1 gin, 1 vermouth and 1 vermouth; a Hanky Panky is 1 gin and 1 vermouth. By volume, the Bronx is 63% fortified wine and bitter, 32% spirit and 5% mixer and juice; the Hanky Panky is 54% fortified wine and bitter and 46% spirit. That makes them close cousins on the ratio map.
The Hanky Panky is the stronger drink in the glass (20.0% ABV against 14.5%) and holds more alcohol (1.59 standard drinks against 1.29). The Hanky Panky has more calories (about 184 against 150) and more sugar (7.1 g against 5.8 g). Both are served in a coupe; the Bronx is shaken, the Hanky Panky stirred.