Make a comparision with "A Rose for Emily," the main character Miss Emily, and Jane: concerning family relationships and social context.
Contrasts between Emily in "A Rose for Emily" and the narrator in "The Yellow Wallpaper" would be easier and more relevant than comparisons, since there aren't many similarities.
The social contexts are worlds apart, literally and figuratively. "Wallpaper" was written primarily to expose what Gilman saw as an asinine and sexist cure for mental illness in women. "Emily" depicts the decaying South after the Civil War.
In terms of family relationships, little is revealed about Emily's father: he was at least somewhat of an aristocrat before the collapse of the economy in the South due to the Civil War, and he kept boys away from Emily, probably because they weren't from backgrounds economically and socially equal to hers.
In contrast, much is revealed about the narrator's husband (a doctor), her brother (a doctor), as well as the then-famous doctor the story was designed to expose and criticize. Medical treatment of mental illness in women was sexist to the core in Gilman's day. Even the term given to her ailment--"hysterical tendency"--is sexist. You might recognize hysterical as related to hysterectomy. It was thought at the time that too much reading, writing, and mental activity would shrink women's ovaries. Mental illness was thought to originate in the ovaries. That kind of thinking is what this story is aimed at.
If you must have similarities, both women in these stories suffer from mental illness, although that even is a weak comparison. Murder and sleeping with a corpse can hardly be compared to post-partum depression and hallucinations.
And both women live in patronizing, sexist societies. Again, though, in terms of family, while the husband/doctor plays a major role in "Wallpaper," sexism comes mostly from the town in "Emily." Too little about Emily's father is revealed to make a comparison with the husband/doctor in "Wallpaper." Sexism is revealed in both stories, but it certainly isn't prevalent in "Emily," while it is the reason for the existence of "Wallpaper."
Incidentally, the narrator is often considered unnamed in "Wallpaper." The use of Jane at the close of the story is ambiguous. The narrator recognizes the man at the door as her husband only seconds earlier, and responds to him as if she were his wife (which she is, of course). She may see herself as the woman escaped from the wall, not as the confined wife, but the use of "Jane" may also suggest something else.
Miss Emily and Jane have one main thing in common, and it is the pressure that they feel from the males in their lives to meet a certain set of expectations which the women, for lack of support, terribly fail to meet.
Miss Emily's father was the looming entity that would define her character and intensify her inner fears. As a man, he is expected to rule over her life a big deal. She was completely dependent on him and, once he was gone, her life became chaotic.
In Jane's case, her life was dependant on her husband and her doctor, both of which could not understand what she was going through they made her life chaotic. They prevented her to take care of herself, just like Emily could not take care of herself due to her father's rulings.
In the end these are two women whose destinies are tied by the influence of men in a forbidding society, This is basically what makes them so similar.