All three are forms of Literature. Plays are the oldest, going back to ancient Greek times. Novels were introduced in the 18th century with Daniel Defoe often credited with having written the first with his most famous work, The Life and Strange Surprizing Adventures of Robinson Crusoe, of York, Mariner, Written by Himself (1719). Short stories came later beginning around the turn of the 19th century. By 1830 several prominent authors like Sir Walter Scott, Dickens and Washington Irving had published collections of short stories, like "Rip Van Winkle." Edgar Allen Poe, Conan Doyle, Pushkin and Gogol added greatly to the short story genre in its early stages. So, one of the first differences is their state of antiquity: plays, novels, short stories.
Plays are meant to be performed on a stage, by actors who have memorized their lines, before a live audience. The action and scope is by necessity limited to what can be produced on a physical stage in a set amount of time. Novels are meant to be read at leisure by one individual or by one individual to a group of listeners. The action and scope is limited only by the logical connections the author has creative power to draw. As we see from modern novels by John Le Carre, action can cover many locations and long time periods. Crime and Punishment by Dostoevsky shows action can cover a compressed period of time, even just days.
Short stories are similar to plays in that the space available necessarily limits the action and the scope. Action and scope must be focused to a narrow time and a single, focused problem, though flashback and flash-forward can expand the time, action and scope as in Faulkner's "A Rose for Miss Emily." So while all three are similar in having action, scope, time, location, and problem (conflict) all differ in how broadly or narrowly these can be developed.
Plays, novels and short stories all differ somewhat in the literary devices they make use of. Of course, structure differs in each with plays being governed by compact acts, novels by wide ranging chapters and short stories by stricter implementation of chapters or their elimination altogether. All develop themes and characters. While all employ foreshadowing and rhetorical techniques, only novels and short stories employ flashbacks and flash-forwards. While all employ symbolism and metaphor, plays address the audience directly. Novels and short stories provide narrated description of setting, characters and events; this is not needed in plays (except minimalist experimental plays) since the audience sees these directly. These are some of the major similarities and differences between plays, novels and short stories.