The difference
Gin, Cointreau and lemon in both. The Breakfast Martini adds a spoon of orange marmalade; the Jasmine adds a little Campari, which turns it pink and bitter-edged, and uses more lemon, so it is tarter and drier.
By the numbers
The Breakfast Martini uses orange marmalade where the Jasmine uses Campari; both have gin, triple sec and lemon juice. In parts, a Breakfast Martini is 3½ gin, 1 triple sec, 1 lemon and ¾ orange marmalade; a Jasmine is 2 gin, ¼ triple sec, ¼ Campari and 1 lemon. By volume, the Breakfast Martini is 54% spirit, 16% liqueur, 16% citrus and 14% syrup and sugar; the Jasmine is 55% spirit, 27% citrus, 9% fortified wine and bitter and 9% liqueur.
The Breakfast Martini is the stronger drink in the glass (17.6% ABV against 16.3%) and holds more alcohol (1.47 standard drinks against 1.28). The Breakfast Martini has more calories (about 182 against 146) and more sugar (8.9 g against 4.3 g). Both are shaken and served in a martini glass.