The difference, by the numbers
Both have simple syrup and lemon juice. Only the New York Sour has rye whiskey, egg white and red wine; only the Whiskey Sour has bourbon. In parts, a New York Sour is 4 whiskey, 1½ syrup, 2 lemon and 1 red wine; a Whiskey Sour is 2¼ whiskey, 1¼ lemon and 1 syrup. By volume, the New York Sour is 47% spirit, 23% citrus, 18% syrup and sugar and 12% mixer and juice; the Whiskey Sour is 50% spirit, 28% citrus and 22% syrup and sugar. That makes them close cousins on the ratio map.
The New York Sour is the stronger drink in the glass (14.2% ABV against 13.4%) and holds more alcohol (1.64 standard drinks against 1.14). The New York Sour has more calories (about 225 against 168) and more sugar (14.7 g against 13.0 g). Both are shaken; the New York Sour is served in a rocks glass, the Whiskey Sour in a coupe.