The main conflict is the difference between the husband's and the wife's WAYS of grieving. The husband is stoic (and was expected to be during this time period) and is not really able to express his grief the way he might want to. The wife is very emotional and is extremely upset that her husband is not expressing himself the way SHE wants him to. What she does not understand is that everyone grieves very differently. She is clearly NOT thinking about this (understandably) as she is threatening to leave her husband over his supposed lack of grief. Frost is able to portray this difference through his use of dialogue, which is genius! As for character, the woman is clearly grieving very openly and does not hold her grief back; whereas, the man is trying to be the "strong one" and he is trying to hold all of it in because this is what was expected of men at the time. They were not encouraged to freely express their emotions and if they did, it was seen as a sign of weakness.
This poem is a dramatic dialogue that takes place between a man and a woman whose child has died. The conflict between the two characters stems from the fact that the woman cannot accept the fact that the man dug the child's grave in a matter-of-fact way. After he dug the grave, he came inside, leaned the spade up against the wall, and spoke to her about the fact that their fence was rotting. The woman cannot forgive what she sees as the man's indifference to their loss.