Since a few days when I play spotify on my chromecast audio, music is not played, but only skipped. Spotify is very busy skipping music, but does not play a thing. This always worked fine. My phone with spotify is a Huawei P8 Lite.
I'm having the exact same problem. Everything worked fine a few days ago, then suddenly, Spotify stopped working with Chromecast Audio. I have two of them set up in a group and casting to either or to the group fails in the same way with Spotify skipping to the next song.
I also have a Chromecast (video) and it's playing along with Spotify perfectly.
I've now updated Spotify on my phone, updated the firmware on my router, rebooted everything and it's still the same result. I've also tried it with Spotify desktop and mobile. Same same.
Other apps can cast to the Chromecast Audio without trouble.
After messaging Spotify Support via @SpotifyCares I went through a number of steps including using a test account on my same setup and it was account specific. They weren't able to fix the account so I ended up creating a new account, they migrated my saves and playlists, and then I was back in business with the new account.
YMMV, but you'll probably need a new account, too.