The difference, by the numbers
They have no ingredient in common: the Hot Buttered Rum has butter and brown sugar, dark rum and water, the Piña Colada white rum, cream of coconut and pineapple juice. In parts, a Hot Buttered Rum is 1 rum and 2½ water; a Piña Colada is 1 rum, ½ coconut and 1 pineapple. By volume, the Hot Buttered Rum is 69% mixer and juice, 28% spirit and 3% syrup and sugar; the Piña Colada is 62% mixer and juice and 38% spirit. Different bottles, similar balance: that makes them close cousins on the ratio map.
The Hot Buttered Rum is the stronger drink in the glass (11.3% ABV against 9.6%) and holds more alcohol (1.35 standard drinks against 1.13). The Piña Colada has more calories (about 267 against 152) and more sugar (23.9 g against 2.6 g). The Hot Buttered Rum is built and served in an Irish coffee glass; the Piña Colada is blended and served in a hurricane glass.