I've always considered Threepenny Opera a musical. It really has the basic elements of a musical including an Overture and finales for each act as well as many songs firmly integrated into the plot.
One of my favorite examples of a play with music is The Grapes of Wrath. The songs and musicians were brilliantly used as backdrop and setting. And though it was categorized as a musical, I always felt The Dead played more as a play with music rather than a musical.
I don't think there is any steadfast cut-and-dry definition that delineates the play with music and musicals. It's really a combination of the amount of music, characters' involvement, the use of the songs, choreography, and the overall tone and intent of the piece. For example, Jersey Boys is very much about the music and the way the songs are performed and featured suggest that the show was intended to be a musical. In The Dead, the songs are performed by the characters, but mostly in meta form where most of the songs themselves have little to do with the plot, but rather abstractly define the characters, the period and the setting.
"What can you expect from a bunch of seitan worshippers?" - Reginald Tresilian