The difference
Gin, both vermouths and orange in both. The Bronx shown here is equal parts gin and vermouths with just a spoon of orange juice; Satan’s Whiskers gives orange juice an equal share and adds Grand Marnier, so it is fruitier, sweeter and a little lighter.
By the numbers
Both have gin, dry vermouth, sweet vermouth, orange juice and orange bitters. Only the Satan’s Whiskers has Grand Marnier. In parts, a Bronx is 1 gin, 1 vermouth and 1 vermouth; a Satan’s Whiskers is 2 vermouth, 2 vermouth, 2 gin, 2 orange and 1 Grand Marnier. By volume, the Bronx is 63% fortified wine and bitter, 32% spirit and 5% mixer and juice; the Satan’s Whiskers is 45% fortified wine and bitter, 22% mixer and juice, 22% spirit and 11% liqueur.
The Bronx is the stronger drink in the glass (14.5% ABV against 13.3%) and holds more alcohol (1.29 standard drinks against 1.08). The Bronx has more calories (about 150 against 139), but the Satan’s Whiskers has more sugar (7.9 g against 5.8 g). Both are shaken and served in a coupe.