Before Hanzo Hasashi died from Sub-Zero's dagger at the beginning of the movie, he told Sub-Zero to "remember this face."
Centuries later when the forces met again near the end of the movie, he said the same thing to Sub-Zero but not in an imperative sentence; he turned it into a question: "Remember this face?"
Hanzo Hasashi conquered the forces in the other world that would have killed him and controlled them instead during his reincarnation. "I'm Scorpion!"
He finally killed Sub-Zero and told his descendant--the main protagonist--to continue the blood line.
But this movie is what that demon guy says before Lord Rayden beamed him away: "death is just a portal." So, the cold Sub-Zero will resurface in another century or just the next generation.
Will there ever be an end to these grudges?
The answer is: no.
So, what do they have to do?
Answer: do the best they can each time they face off.