The difference
Gin and sweet vermouth stirred in both. The Bijou adds an equal third of green Chartreuse, which makes it more herbal, sweeter and stronger; the Martinez sweetens with just a spoon of maraschino and a little more orange bitters, so the gin and vermouth lead.
By the numbers
The Bijou uses Green Chartreuse where the Martinez uses Maraschino liqueur; both have gin, sweet vermouth and orange bitters. In parts, a Bijou is 1 gin, 1 Chartreuse and 1 vermouth; a Martinez is 1 gin and 1 vermouth. By volume, the Bijou is 34% liqueur, 33% spirit and 33% fortified wine and bitter; the Martinez is 48% spirit, 47% fortified wine and bitter and 5% liqueur.
The Bijou is the stronger drink in the glass (24.4% ABV against 19.7%) and holds more alcohol (1.89 standard drinks against 1.55). The Bijou has more calories (about 234 against 184) and more sugar (12.0 g against 8.0 g). Both are stirred and served in a coupe.