The difference, by the numbers
Both have sweet vermouth. Only the De La Louisiane has rye whiskey, Bénédictine, absinthe and Peychaud’s bitters; only the Tipperary has Irish whiskey, Green Chartreuse and Angostura bitters. In parts, a De La Louisiane is 1 whiskey, 1 vermouth and 1 Bénédictine; a Tipperary is 2 whiskey, 1 vermouth and ½ Chartreuse. By volume, the De La Louisiane is 36% fortified wine and bitter, 32% liqueur and 32% spirit; the Tipperary is 55% spirit, 28% fortified wine and bitter and 17% liqueur. That makes them close cousins on the ratio map.
The Tipperary is the stronger drink in the glass (23.9% ABV against 23.1%) and holds more alcohol (1.86 standard drinks against 1.41). The Tipperary has more calories (about 212 against 179), but the De La Louisiane has more sugar (10.1 g against 7.5 g). Both are stirred; the De La Louisiane is served in a coupe, the Tipperary in a martini glass.