The difference lies in two “imitations of an action”, according to Aristotle. Drama, one of three poetic structures (epic and lyric being the others), is “imitation of an action by language without narrator.” That is, unlike epic, which has multiple narrators (the omniscient narrator plus various characters—Odysseus, Poseidon, etc.—who speak), and lyric (which has one narrator –“I wandered lonely as a cloud”), drama has no narrator (what looks like exceptions—Ben Jonson’s inductions, or Glass Menagerie’s Tom Wingfield, etc.—are simply characters speaking to the audience.) Theatre, on the other hand, is “imitation of an action by means of action”; that is, the actions that tell the story are acted out in three dimensions (this includes the dialogue, which modern scholars call “speech-acts”), with blocking, proxemics, facial expressions, costumes, etc.
An example ready to hand is Hamlet’s first entrance at court. The theatre-goer, watching Hamlet the theatrical performance, sees an imitation of an action in Hamlet’s black robes, inappropriate for this court gathering. The reader of Hamlet the drama, however, knows only that Hamlet is among the characters on stage at the moment, because his name is listed in the (written) stage direction. When Gertrude says, “Hamlet, cast thy nighted colors off” the (naïve) theatre-goer sees her addressing the figure in black and therefore knows that that actor is “imitating” that character, Hamlet. The reader of the drama, however, must perform the reading act called recursive reading, and re-dress the figure of Hamlet he or she has been constructing in the mind (for that is what reading is). Taking this small example we can see what the difference is between “reading” the dramatic script and “reading” the theatrical “imitation in the form of action.” In fact, the dramatic script can be considered a recipe, to be mixed and “cooked” on the stage. If a food recipe could be read as literature, it would be a drama. All the tools of literature—metaphor, meter, rhyme, etc., etc.--are available to the playwright, so the reader gets a literary experience from the text even though that is not its primary function. There is even a subgenre of drama, called literary plays (Seneca’s plays may have been of this type) (not to be confused with reader’s theatre, which is the reading of a playscript as a performance for an audience).