Its pretty hard to beat Spotify unless you have some specific taste in music they dont cover well. But overall I have a hard time seing a better service than theirs.
It just works, is fast and has a large db of content.
I've tried both and have stuck with Rdio. The social aspect is superior. I love being able to see playlists and recommendations from random people beyond my Facebook friends. Spotify's desktop app is definitely faster, though I find Rdio to be better designed.
But there is a BIG difference here if we're talking about the free services of both. Spotify is always free, and rdio is only free for 6 months. Yes it's probably a good thing to pay for a service that you like (I do), but some people don't, and that's going to be where they prefer Spotify.
When you don't pay for it, Spotify interrupts your music with shitty commercials for country music, while Rdio just cuts you off after awhile. Hence, Rdio was the one I ended up using long enough to decide to subscribe.
It just works, is fast and has a large db of content.