The difference, by the numbers
Both have simple syrup and sparkling wine. Only the Old Cuban has mint, aged rum, lime juice and Angostura bitters; only the Russian Spring Punch has vodka, lemon juice and crème de cassis. In parts, an Old Cuban is 1½ rum, ¾ lime, 1 syrup and 2 bubbles; a Russian Spring Punch is 1¾ vodka, 1¾ lemon, 1 cassis, ¾ syrup and 4 bubbles. By volume, the Old Cuban is 38% mixer and juice, 29% spirit, 19% syrup and sugar and 14% citrus; the Russian Spring Punch is 44% mixer and juice, 19% spirit, 19% citrus, 11% liqueur and 7% syrup and sugar. That makes them close cousins on the ratio map.
The Old Cuban is the stronger drink in the glass (12.1% ABV against 11.3%) and holds more alcohol (1.46 standard drinks against 1.10). The Old Cuban has more calories (about 225 against 167) and more sugar (19.6 g against 13.5 g). Both are shaken; the Old Cuban is served in a coupe, the Russian Spring Punch in a highball.