I found this very interesting as to how all of this came about. There is sample sounds of the first one he turned in as well as the final piece. So we can see the difference. I do suggest that you read the whole write up to get a real understanding. If for anything the very last part he wrote brought me to tears.
"Bella's Lullaby," as it appears in this film, was not written to be a lullaby but to speak of love - ecstatic, tormented love. Here's that story.
Years ago I was in love with an amazing and challenging woman named Christine Sciulli. She left me, I was heartbroken, and I wrote a piece of music that tried to express the thrill and pain of having my heart pierced. She wouldn't speak to me, so I sent her the music to speak in my place.
Years later (April 2008 to be exact) I came to Oregon to meet Catherine Hardwicke and see some of the film she was shooting, Twilight. She mentioned that the producing company, Summit Entertainment, had just requested that a new scene be added to the film. The scene existed in the novel but hadn't originally been part of the screenplay (films are not one-to-one translations of books). In this scene Edward would play piano for Bella. I wasn't officially working on the film at that point and they didn't know what Rob Pattinson (Edward) should play during the shoot. This is not an unusual situation - many films have an actor sitting at a keyboard, swaying back and forth, pretending to play music that was only written after the film has been shot and edited. This case was unusual in that Rob is a fine musician, and fully capable of playing the piano or probably any other instrument.
REST CAN BE FOUND HERE