The line between these two words is becoming blurred because both words are hardly ever in common use nowadays.
Troupe refers to a travelling group of minstrels who used to visit castles to entertain. Later, when Burlesque was popular, a troupe was a group of actors/entertainers who sang & danced & moved from town to town.
A troop refers to a grup of soldiers who serve togther under a commander.
When one is thought to be a trouper, it means the show must go on, even if sickness or injury might indicate otherwise. Someone may be a trouper if they carry on under any kind of adversity.
A trooper is one individual in a troop of soldiers. It doesn’t indicate someone carrying on under adversity. thus a trooper might be a trouper if he continues his duties even while injured.