The most obvious would be the characters of Miss Emily and Granny Weatherall. Both are very stubborn old women who are set in their ways and ho have jaded views of reality. Both harbor resentments about their pasts, too. Both women have also been betrayed and jilted. Emily, for example, has been betrayed by her father because of the way he raised her. He did not let her have a normal childhood and adolescence because he kept her so sheltered. Granny Weatherall was jilted by George, who left her at the altar, and by Hapsy, her child who died. There are many more similiarities and quite a few differences, too!
Another similarity is the setting. Both stories are set in Southern states. Both women also have similar value systems that are clearly Southern in nature.
Some of the differences include the authors' writing styles, the themes of the stories, and the tones of each story.
If you are writing for an essay, you might want to check out eNotes' free guide called How to Write a Compare-and-Contrast Essay.