The conflict in the short story Lather and Nothing Else is man vs. himself, the main character, the barber is secretly a member of the resistance that is fighting against the government in place. .
The conflict comes into play when a captain of the enemy comes into the main characters barbershop to get a shave, which gives the barber a perfect opportunity to kill him and become a hero for the resistance, or to be looked upon as a coward by the enemy. If he took this opportunity, it would strengthen the resistance and weaken the government by eliminating one of there ruthless captains, who hunts down members of the resistance and puts there punishments up for display so that all will see it. Seeing people mutilated, killed, or tortured is supposed to discourage people from acting out against the government, and this is the punishment for members of the resistance. If the main character does kill the captain it will be the end of life as he knows it. He will no longer own a barbershop or be able to live out his life normally anymore, because he will be spending his time fleeing or hiding from the government.
He could escape to another country or the government would forget about him, but he would still have to live with the fact that he killed another man and this would haunt him for the rest of his days. .
In the end, I think he made the right choice by not killing the captain because to do so would lower himself to the captains level.