I'm not sure I can explain it exactly, but here are my thoughts. FOR THE BETTER describes the change, it doesn't necessarily say what we changed into. It just means 'we improved' in some way. "We changed the wallpaper to yellow" is more specific. It is saying what the wallpaper changed. You could use a similar construction for the first sentence too if it was more specific, ex: The house changed for the better. OR After we cleaned, painted and did repairs, the house changed from a depressing mess to a beautiful place to live.