Having another character to interact with in a dialogue can help flesh out another character. It also gives answers to an audience for questions they might be asking. In Orphans, Treat revels in the purchasing power of Harold's American Express card, which causes Harold to say, "You're developing a sense of style, Treat, that's fine...but remember, please, everything in moderation." Treat replies that he doesn't know much about moderation, to which Harold responds, "I can see that, Treat." From these lines, we can deduce that Treat has probably never been able to buy the sorts of clothes he has been buying, and he is very naive about how money and consumption work.
Whereas monologues are fairly straightforward in their narrative, dialogues allow for more of a dramatic build. In Angels in America, Harper confronts Joe about his sexuality. It would be boring if the interaction consisted of nothing more of, "Are you gay?", "Yes/no." Instead, Harper anxiously has to try and pull his confession out. Joe denies that he is a homosexual, and then deflects her question by stating that he is "a very good man who has worked very hard to become good" and accusing her of wanting to destroy his image. In dialogue, the verbal back-and-forth gives the audience a much more complex narrative.
Dialogues also provide a chance for juxtaposition. In Closer, the audience observes two couples at once--Dan and Alice, and Anna and Larry. Dan has cheated on Alice, and Anna has done the same to Larry. While both sets of dialogue start out fairly the same, they come to very different ends. Alice, while still angry, is mostly in grief over Dan's decision to leave her. Larry, on the other hand, is much more furious and distraught over Alice's actions.