The first way a play communicates with an audience differently than other forms of literature (such as poetry, short stories, and novels) can be found in your question: literary works generally only have an audience of one, whereas a play has an audience of many. While reading literature affords a reader time to contemplate what he is reading and the ability to reread for clarity or pleasure, watching a play becomes a collective exercise. Reactions are often contagious, as you may have experienced in a particularly funny or emotional movie, for example.
Another way experiencing a play is different than reading a piece of literature, of course, the ability to use other senses to understand and appreciate the work. An audience does not rely on the written word but is able to see and hear the story unfold.
While a reader is able to (and must) create his own interpretation of the literary work, audience members at a play are given the director's and actors' interpretations of the work. In this case, the written literary work is more interactive; however, a play may present a point of view, perspective, or interpretation which would not have surfaced through reading.
Finally, an audience has the power to change the play, whereas a reader is powerless to do anything but accept the words as written. A lively audience can give a kind of energy to a performance; while the words and actions of the play may not change, it is true that an engaged audience creates a more engaged and even interactive performance.
Plays (technically the actors who speak the words of the plays) literally communicate with their audiences; literary works offer words which the readers must absorb or skim through--or ignore completely. Both plays and literary works are open to interpretation, but the experience is entirely different in each case. There is value in gleaning themes by reading rather than having them demonstrated on stage; and there is value in hearing written words come to life.
Reading good literature is an experience that has value; it is different than seeing a story come to life on stage, but it is no less valuable.