The difference, by the numbers
Both have sparkling wine. Only the Champagne Cocktail has sugar, Angostura bitters and Cognac; only the Hugo Spritz has lemon balm syrup, soda water and mint. In parts, a Champagne Cocktail is 1 sugar, 1½ brandy and 14 bubbles; a Hugo Spritz is 8 bubbles, 1 lemon balm syrup and 1½ soda. By volume, the Champagne Cocktail is 85% mixer and juice, 9% spirit and 6% syrup and sugar; the Hugo Spritz is 90% mixer and juice and 10% syrup and sugar. That makes them close cousins on the ratio map.
The Champagne Cocktail is the stronger drink in the glass (15.1% ABV against 7.2%), but the Hugo Spritz is the bigger drink, so it holds more alcohol (1.01 standard drinks against 0.90). The Hugo Spritz has more calories (about 147 against 112) and more sugar (10.8 g against 5.3 g). Both are built; the Champagne Cocktail is served in a flute, the Hugo Spritz in a wine glass.