It's mostly a question of the amount of emphasis on music and the degree to which music is used to tell the story. In certain plays about musicians, (PAUL ROBESON, PIAF, LADY DAY AT EMERSON'S BAR & GRILL about Billie Holiday, DINAH WAS about singer Dinah Washington) or in one recent example, a "non-musician" who thought she was one (SOUVENIR about socialite and legendary bad singer Florence Foster Jenkins), the subject's musical ability (lack thereof) is part of their story so the playwright uses examples of them performing to give the audience a fuller portrait of them as artists and people. In Nora Ephron's IMAGINARY FRIENDS which starred Cherry Jones and Swoosie Kurtz a few seasons ago, the play used a chorus of elegantly dressed dancer-singers as a sort of Greek chorus to comment on the action.
In each of these plays, the music is secondary to the main drama and simply used to flesh out the main character(s) and to tell the audience about the essense of the character(s) that mere words couldn't. In a true musical, the music has at least as important a function in the overall work as the book and is typically actually more important than the book in many ways. In a play with music, the music isn't truly essential and the play could work without it.
"What a story........ everything but the bloodhounds snappin' at her rear end." -- Birdie
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"The Devil Be Hittin' Me" -- Whitney