The difference
Both are New Orleans drinks of rye, sweet vermouth, Bénédictine and Peychaud’s. The Vieux Carré adds Cognac and uses only a spoonful of Bénédictine; the De La Louisiane makes Bénédictine an equal part and adds absinthe, so it is sweeter and more herbal.
By the numbers
The De La Louisiane uses absinthe where the Vieux Carré uses Cognac; both have rye whiskey, sweet vermouth, Bénédictine and Peychaud’s bitters. In parts, a De La Louisiane is 1 whiskey, 1 vermouth and 1 Bénédictine; a Vieux Carré is 1 whiskey, 1 brandy and 1 vermouth. By volume, the De La Louisiane is 36% fortified wine and bitter, 32% liqueur and 32% spirit; the Vieux Carré is 63% spirit, 32% fortified wine and bitter and 5% liqueur.
They are about equally strong in the glass (23.1% and 22.8% ABV), but the Vieux Carré is the bigger drink, so it holds more alcohol (1.85 standard drinks against 1.41). The Vieux Carré has more calories (about 205 against 179), but the De La Louisiane has more sugar (10.1 g against 6.0 g). Both are stirred and served in a coupe.