The difference, by the numbers
Both have sweet vermouth. Only the Bitter Giuseppe has Cynar, lemon juice and orange bitters; only the Negroni Sbagliato has Campari and sparkling wine. In parts, a Bitter Giuseppe is 2 Cynar, 1 vermouth and ¼ lemon; a Negroni Sbagliato is 1 Campari, 1 vermouth and 1 bubbles. By volume, the Bitter Giuseppe is 92% fortified wine and bitter and 8% citrus; the Negroni Sbagliato is 67% fortified wine and bitter and 33% mixer and juice. That makes them close cousins on the ratio map.
The Negroni Sbagliato is the stronger drink in the glass (14.9% ABV against 12.1%), but being smaller it holds about the same alcohol (0.94 and 0.88 standard drinks). The Bitter Giuseppe has more calories (about 147 against 135) and more sugar (13.7 g against 12.1 g). The Bitter Giuseppe is stirred and served in a rocks glass; the Negroni Sbagliato is built and served in a goblet.