You are probably missing some libs, I don't have control over CentOS. There are no success stories published on CentOS either, so I just presume there are missing bits. Basically, let's say CentOS isn't supported by the RPM (there is more Fedora-specific stuff in that specfile...)
I suggest you try the installer at https://github.com/leamas/spotify-make . This will give you more control and ways to walk around possible problems then a shrink-wrapped, fedora-specific RPM.
I tried your installer successfully (listening to music right now 🙂 )
However I get this right at startup:
/usr/local/lib/spotify-client/spotify: /usr/local/lib/spotify-client/libcrypto.so.0.9.8: no version information available (required by /usr/local/lib/spotify-client/spotify) /usr/local/lib/spotify-client/spotify: /usr/lib64/libssl.so.0.9.8: no version information available (required by /usr/local/lib/spotify-client/spotify)