The similarities are in the period, the limited setting (all happens in one location in both works—in a home/house), symbolic issues related to the setting, a triggering event around illness or medical intervention in the family, and, especially, the focus on female subjugation, and the desire/attempt to break free. There are several differences between these two works. The most basic is form: one is a play, and one a story told by the main character. This means viewers get what Nora externalizes—what she says and does—while the narrator of Gilman's story shares thoughts as well. After that, the nature of their rebellion. Nora is emotional at the play's start, and plays at being mindless. The woman in "The Yellow Wallpaper" starts rational, and ends up mad.
In both the short story "The Yellow Wallpaper" and the play A Doll's House, we have a Victorian-era female protagonist who is trapped in some way. For the narrator of "Yellow Wallpaper," she is more literally trapped in a room where she is supposed to be taking the "rest cure" for depression. For Nora of A Doll's House, she is trapped into a prescriptive feminine role that requires her to act submissive and almost childish while deceiving her husband at the same time.
The main difference between the two stories is that the short story's narrator descends into madness. Nora, on the other hand, "awakens" to her situation and decides to leave her husband and start a new life.