The difference, by the numbers
Both have sweet vermouth. Only the Bijou has gin, Green Chartreuse and orange bitters; only the De La Louisiane has rye whiskey, Bénédictine, absinthe and Peychaud’s bitters. In parts, a Bijou is 1 gin, 1 Chartreuse and 1 vermouth; a De La Louisiane is 1 whiskey, 1 vermouth and 1 Bénédictine. By volume, the Bijou is 34% liqueur, 33% spirit and 33% fortified wine and bitter; the De La Louisiane is 36% fortified wine and bitter, 32% liqueur and 32% spirit. That makes them close cousins on the ratio map.
The Bijou is the stronger drink in the glass (24.4% ABV against 23.1%) and holds more alcohol (1.89 standard drinks against 1.41). The Bijou has more calories (about 234 against 179) and more sugar (12.0 g against 10.1 g). Both are stirred and served in a coupe.