The difference, by the numbers
Both have soda water. Only the Rome with a View has Campari, dry vermouth, lime juice and simple syrup; only the Spritz has sparkling wine and Aperol. In parts, a Rome with a View is 1 Campari, 1 vermouth, 1 lime, ¾ syrup and 2 soda; a Spritz is 3 bubbles, 2 Aperol and 1 soda. By volume, the Rome with a View is 35% fortified wine and bitter, 35% mixer and juice, 17% citrus and 13% syrup and sugar; the Spritz is 67% mixer and juice and 33% fortified wine and bitter. That makes them close cousins on the ratio map.
The Spritz is the stronger drink in the glass (8.9% ABV against 5.7%) and holds more alcohol (0.98 standard drinks against 0.71). Calories are close (about 163 and 154), but the Rome with a View has more sugar (22.5 g against 13.7 g). The Rome with a View is shaken and served in a highball; the Spritz is built and served in a wine glass.