Well, compared to writing a screenplay, a stage play is almost the exact opposite. The rule in screenwriting is "show it, don't say it." Plays, on the other hand, are almost entirely dialogue-driven since it's difficult to stage very much beyond that -- unless you're a big-budget Broadway musical. The genres for stage plays are more narrowly defined as well. Mysteries, comedies and dramas work well; not so much sci-fi, fantasy, action, etc. You need strong characters and a good ear for dialogue in both mediums, but with a stage play you need to be able to engage your audience for the entire duration of the production using ONLY those tools, so they'd better be solid.
Regarding music, obtaining the rights to perform copyrighted work is the same as for film: contact the artist's rep and/or the music publisher and either pay a fee or get something in writing giving you permission. As in film, generally that's the responsibility of the play's producer, although if you as the writer want to reproduce someone else's lyrics in your script and you expect to have it published, you will need to obtain permission as well. This is not recommended if you can at all avoid it, as it can be time-consuming and expensive, and could potentially dissuade a producer from mounting your show.
