One of the major similarities between the two stories is the way that there is a shocking revelation at the end of the story that significantly changes the outcome or the atmosphere. In The Story of an Hour, it is the entrance of the husband and the immediate death of the wife, in War it is the question from the woman about the other man's son, is he in fact dead. In the latter, it is this question that finally breaks through the man's facade of not needing to grieve and he breaks down sobbing and wailing.
Both stories also raise questions about separation and loyalty and love that are explored by the characters.
They are of course very different in their subject matter and the fact that in War, it is the possibility of death that is haunting to the one couple and a past death that breaks down the other man. In The Story of an Hour, it is an assumed death that turns out to be false that then leads to the death of the wife.