The difference, by the numbers
Both have water. Only the Hot Buttered Rum has butter and brown sugar and dark rum; only the Hot Toddy has bourbon and honey. In parts, a Hot Buttered Rum is 1 rum and 2½ water; a Hot Toddy is 2 whiskey, 1 honey and 5½ water. By volume, the Hot Buttered Rum is 69% mixer and juice, 28% spirit and 3% syrup and sugar; the Hot Toddy is 64% mixer and juice, 24% spirit and 12% syrup and sugar. That makes them close cousins on the ratio map.
The Hot Buttered Rum is the stronger drink in the glass (11.3% ABV against 10.8%) and holds more alcohol (1.35 standard drinks against 1.14). The Hot Toddy has more calories (about 209 against 152) and more sugar (26.2 g against 2.6 g). Both are built and served in an Irish coffee glass.