The difference, by the numbers
Both have sugar. Only the Champagne Cocktail has Angostura bitters, Cognac and sparkling wine; only the Mulled Wine has red wine, cinnamon sticks and cloves. In parts, a Champagne Cocktail is 1 sugar, 1½ brandy and 14 bubbles; a Mulled Wine is 4½ red wine and 1 sugar. By volume, the Champagne Cocktail is 85% mixer and juice, 9% spirit and 6% syrup and sugar; the Mulled Wine is 82% mixer and juice and 18% 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 12.5%), but the Mulled Wine is the bigger drink, so it holds more alcohol (1.43 standard drinks against 0.90). The Mulled Wine has more calories (about 256 against 112) and more sugar (26.1 g against 5.3 g). Both are built; the Champagne Cocktail is served in a flute, the Mulled Wine in an Irish coffee glass.