The difference, by the numbers
Both have lime juice. Only the Missionary’s Downfall has white rum, peach liqueur, honey syrup, mint and fresh pineapple; only the Singapore Sling has gin, cherry liqueur, triple sec, Bénédictine, pineapple juice, grenadine and Angostura bitters. In parts, a Missionary’s Downfall is 1 rum, ½ peach, ½ lime, 1 honey and 2¾ fresh pineapple; a Singapore Sling is 1 gin, ½ cherry, 4 pineapple, ½ lime and ¼ grenadine. By volume, the Missionary’s Downfall is 49% mixer and juice, 17% spirit, 17% syrup and sugar, 9% liqueur and 8% citrus; the Singapore Sling is 58% mixer and juice, 15% spirit, 15% liqueur, 7% citrus and 5% syrup and sugar. That makes them close cousins on the ratio map.
The Missionary’s Downfall is the stronger drink in the glass (10.8% ABV against 7.7%), but the Singapore Sling is the bigger drink, so it holds more alcohol (1.27 standard drinks against 0.88). The Singapore Sling has more calories (about 265 against 196), with about the same sugar (26.7 g and 27.6 g). The Missionary’s Downfall is blended and served in a goblet; the Singapore Sling is shaken and served in a hurricane glass.