The differences are entirely in the telling; the story itself can and has been adapted to a play several times. This allows what might be called the most important similarity; the story is confined to the house of the White family, so in adaptation it needs only a single location. The story itself, though, is told as a straightforward third-person omniscient narrative. In a play, the stage directions would replace narration; actors would have more leeway in interpreting their own characters, both in motivation and emotion. In the story, the narration explains most of their motivations, actions, and emotions directly; the reader understands the story through its telling on the page, instead of through the interpretation of the adaptors.
He sat alone in the darkness, gazing at the dying fire, and seeing faces in it. The last face was so horrible and so simian that he gazed at it in amazement. It got so vivid that, with a little uneasy laugh, he felt on the table for a glass containing a little water to throw over it. His hand grasped the monkey's paw...
(Jacobs, "The Monkey's Paw," gaslight.mtroyal.ca)
This scene, for example, would need to be expanded through inner monologue; fire cannot be controlled on stage to form faces, so Mr. White would need to comment on the fact and show, through acting, how it affects him. In the story, it is explained as a matter of fact; no further exposition is necessary. This is one of the major differences between literature prose and plays.