I was listening to NPR today and heard a piece on Christoph Eschenbach, currently the conductor of the National Symphony Orchestra (pictured right). Indirectly, he gave what I found to be a very cogent answer to this question:
"Music can invade you totally, and can speak all the languages of emotion to you. You can express all the language of emotion with that music."
We writers often complain about not being able to fully trasmit our emotions. Music, it seems, can fill this gap which writing cannot. To say that writing is just the language of the intellect and music is just the language of emotions sets up a false dichotomy, but it also gives a good sense of their respective strengths.
Update (2hrs later): When I wrote this post, I didn't consider poetry, arguably the most emotional form of writing. Even still, I feel music has a superior fidelity for emotions. Music doesn't rely on words, or even concepts, and that allows it to speak to us from a place in our minds before words or concepts even form.