Spotify can be a but clunky on the Watch until you get used to it. And even then...
The most common problem seems to be that the AirPods and Spotify both prefer to use the iPhone, rather than the Watch. Best thing to do is:
* Restart both devices.
* Open and close Spotify on the iPhone (to make sure it's closed).
* Disable Bluetooth on the phone (form Settings, not from the Control Centre).
* Start the music playing from the watch and it should prompt you to connect to your AirPods.
When the music is playing from the watch you should see it shown in the bottom right corner.
Playing on Phone:
Playing on Watch:
Usually I'm playing from the watch only if I'm not near my phone (out for a run) and therefore it's usually a FAR smoother process. I don't need to then bother about disabling bluetooth, closing Spotify etc. But when the two devices are near then it will typically have the phone trump the watch.
The other PITA is downloading songs to the watch. By default the watch will download using the phone's Internet connection via BT. It's far quicker to download over Wifi, or even cellular. Just disable the BT on the phone as above (Settings, not Control Centre) to force it to avoid BT.