There's not really that much difference I guess. A lot of song writers write the music before the words though. I don't know if you would know who this is. The Doors were a band in the 1980's. Jim Morrison was the lead singer. His lyrics were mostly poems first.
You make an excellent point and ask an excellent question. In my opinion, poetry, even in the West, can and should be read aloud (and sung aloud, too, I suppose -but I think I'll try that in the shower first before inflicting the experience on friends and family ). Poetry is about words.
Songwriters, though, generally compose some music to go with their words (or someone else's words -the lyricist's). The poetry that goes with the music in songs is generally of the more structured and lyrical kind. Western poetry, on the other hand comprises a lot of variety, and some may be less suited to music than others. I've yet to see a limerick set to music, but it could happen!