The difference, by the numbers
Both have pineapple juice. Only the Jungle Bird has dark rum, Campari, lime juice and Demerara syrup; only the Pisco Punch has pisco, simple syrup, lemon juice, dry white wine and cloves. In parts, a Jungle Bird is 3 rum, 1½ Campari, 3 pineapple, 1 lime and 1 syrup; a Pisco Punch is 4 pisco, 1½ pineapple, 1 syrup, 1 lemon and 2 wine. By volume, the Jungle Bird is 32% mixer and juice, 32% spirit, 16% fortified wine and bitter, 10% syrup and sugar and 10% citrus; the Pisco Punch is 42% spirit, 37% mixer and juice, 11% syrup and sugar and 10% citrus. That makes them close cousins on the ratio map.
The Pisco Punch is the stronger drink in the glass (13.2% ABV against 10.9%) and holds more alcohol (1.56 standard drinks against 1.32). Calories are close (about 216 and 207), but the Jungle Bird has more sugar (19.7 g against 12.2 g). Both are shaken; the Jungle Bird is served in a rocks glass, the Pisco Punch in a goblet.