The mother and father were in a very difficult situation; they were trying to survive in a brutal world filled with barbaric people, and they had a little baby to take care of.  Under such stress, conflict and anxiety are inevitable.  The main issue that they argue over is whether or not that life was even worth living.  The mother lives in constant anxiety over what she feels is the inevitable brutalization of all of them at the hands of the barbarians.  Every day she wakes up and faces the fact that most likely, as she states in the novel,

"Sooner or later they will catch us and they will kill us.  They will rape me.  They'll rape him.  They are going to rape us and kill us and eat us and you won't face it."

She states that she does not want to live long enough to see that happen to all of them.  They argue because she wants to leave the world; she is finished.  She wants it over and done with.  She says that there is nothing to live for anymore, and that death has more to offer than this world does.  He wants her to stay; he is afraid that he can't survive without her.  He doesn't want to be alone.  He begs her to stay with them, and she refuses.  She does eventually leave, taking her own life instead of living any longer in that one.  It's a sad situation, one that is understandable and heart-wrenching all at the same time.  I hope that helped; good luck!