The difference, by the numbers
Both have Green Chartreuse and sweet vermouth. Only the Bijou has gin and orange bitters; only the Tipperary has Irish whiskey and Angostura bitters. In parts, a Bijou is 1 gin, 1 Chartreuse and 1 vermouth; a Tipperary is 2 whiskey, 1 vermouth and ½ Chartreuse. By volume, the Bijou is 34% liqueur, 33% spirit and 33% fortified wine and bitter; the Tipperary is 55% spirit, 28% fortified wine and bitter and 17% liqueur. That makes them close cousins on the ratio map.
The Bijou is the stronger drink in the glass (24.4% ABV against 23.9%), but being smaller it holds about the same alcohol (1.89 and 1.86 standard drinks). The Bijou has more calories (about 234 against 212) and more sugar (12.0 g against 7.5 g). Both are stirred; the Bijou is served in a coupe, the Tipperary in a martini glass.