The first time I heard New Beginning I had no idea which songs were the singles and which songs were hits. So I heard the album and Stay became my favorite song. In true 'me' fashion, it was his last and lowest-charting single.
The version I knew of before I went YouTube-hopping for his stuff today was the one on the album and that was much more piano/guitar-based with less 2000-ish pop elements. It sounded a lot older as well than the single version, which I didn't like the first time I heard(about on hour ago?) but now I think it's pure and utter brilliance.
So let's talk about the album version first. As I said a while ago, it's more piano/guitar-based and the vocals are smoother - they sound like the song came straight out of a studio, which it did. Towards the end the vocals are all heavily layered and you can hardly make out Stephen's voice anymore. Still, amazing melody and this version has it's quirks.
The single/video version is much more 2000 pop, the arrangement sounds like ...Baby One More Time. His vocals are a lot more raw as well, they layering on the album version is gone and it's just him singing with minimal backing during the choruses. One reason why I think they made the song like this was for the dance routine. The album version was nice, but a dance routine didn't go very well with it. At least not as much as this one does.
Either version, I LOVE THE SONG. Brilliant melody, great arrangement.