I've been using the Android app for a bit. To be honest, the UI feels a bit dated and doesn't really fit the Holo design guidelines which you can find here. One example of a really nice redesign is attached to this post.
Thanks for considering this.
EDIT: Now that Lollipop has been released along with the Material Design Guidelines, I think it would be a good idea to update the UI to fit these guidelines. A nice redesign is now attached to this post.
Updated on 2017-08-29
Hey everyone. This idea has been marked as "Not Right Now" since 2014 and today we're marking it as "Case Closed".
To ensure Spotify feels familiar across all the different platforms and moments in peoples lives we work hard to ensure the experience feels like Spotify everywhere first, which means we follow our UI guidelines and would not follow the material design guidelines like for like.
However we are always sensitive to the nuances of the platform guidelines and work hard to balance the familiarity of the platform with the familiarity of Spotify.
We appreciate all your feedback, but we want to be transparent about the fact that this won't be fully implemented. Keep your eye on the Ideas Board for other announcements about idea we can implement for all of you.