The difference, by the numbers
Both have dark rum. Only the Hot Buttered Rum has butter and brown sugar and water; only the Painkiller has pineapple juice, orange juice and cream of coconut. In parts, a Hot Buttered Rum is 1 rum and 2½ water; a Painkiller is 2 rum, 4 pineapple, 1 orange and 1 coconut. By volume, the Hot Buttered Rum is 69% mixer and juice, 28% spirit and 3% syrup and sugar; the Painkiller is 75% mixer and juice and 25% spirit. That makes them close cousins on the ratio map.
The Hot Buttered Rum is the stronger drink in the glass (11.3% ABV against 7.1%), but being smaller it holds about the same alcohol (1.35 standard drinks each). The Painkiller has more calories (about 342 against 152) and more sugar (34.0 g against 2.6 g). The Hot Buttered Rum is built and served in an Irish coffee glass; the Painkiller is shaken and served in a goblet.