I solved the issue by removing 0.0.0.0 redirector.gvt1.com from my hosts file. This entry blocks Firefox from updating the Widevine Content Decryption Module provided by Google.
If this doesn't immediately fix the problem for you, untick 'Play DRM-controlled content' under 'General'. This will force the uninstall of the DRM decryption module. Tick it again and visit Spotify to download the DRM module once again.