The difference, by the numbers
Both have orange bitters. Only the Bronx has gin, dry vermouth, sweet vermouth and orange juice; only the Trident has dry sherry, Cynar, aquavit and peach bitters. In parts, a Bronx is 1 gin, 1 vermouth and 1 vermouth; a Trident is 1 dry sherry, 1 Cynar and 1 aquavit. By volume, the Bronx is 63% fortified wine and bitter, 32% spirit and 5% mixer and juice; the Trident is 67% fortified wine and bitter and 33% spirit. That makes them close cousins on the ratio map.
The Trident is the stronger drink in the glass (17.6% ABV against 14.5%), but being smaller it holds about the same alcohol (1.29 and 1.28 standard drinks). Calories are close (about 150 and 144), but the Bronx has more sugar (5.8 g against 4.6 g). Both are served in a coupe; the Bronx is shaken, the Trident stirred.