Can’t play Spotify through Bluetooth headphones via Apple Watch Series 5

Officially, you cannot. Spotify does not support the Apple Watch as a device to play its music. The ONLY thing Spotify allows is for the Apple Watch to be a remote control for the Spotify iPhone app. This is all Spotify's decision.

Pandora and Apple Music fully support the Apple Watch with music streaming, offline music, and Siri integration.

There are unofficial hacks to get Spotify music onto the Apple Watch for offline play

https://www.wareable.com/apple/how-to-use-spotify-on-apple-watch-6609

https://www.ondesoft.com/spotify-converter-for-mac/play-spotify-music-on-apple-watch-without-iphone.html 

NOTE: One of the articles is slightly out of date, as watchOS 6 does allow 3rd party streaming; Pandora is offering Apple Watch LTE streaming, off-line playlists and Siri integration on the Apple Watch.  Plus Apple Music supports the Apple Watch.

Listen to music, podcasts, and audiobooks on your Apple Watch

https://support.apple.com/en-us/HT204691

Add music to Apple Watch

https://support.apple.com/guide/watch/add-music-apd483798d11/watchos

Play music on Apple Watch

https://support.apple.com/guide/watch/play-music-apd70768b20b/watchos

Apple Music

https://support.apple.com/en-us/HT204691

https://support.apple.com/guide/watch/play-music-apd70768b20b/watchos

Pandora Streaming

https://appleinsider.com/articles/20/02/14/pandora-watch-app-no-longer-requires-the-use-of-an-iphone-to-stream-music

Hello Katelyn1129,

It sounds like the Spotify application on your Watch is not outputting sound to your Bluetooth Headphones.

Here's a resource that can help:

Use AirPods and other Bluetooth accessories with Apple Watch

Specifically, use these steps:

1. "If you see  on your watch face, Airplane Mode is on. On your Apple Watch, tap Settings > Airplane Mode, then turn it off. 
2. Make sure your accessory is fully charged and powered on.
3. If you see the accessory in Settings > Bluetooth on Apple Watch, but you can’t connect, use the steps above to unpair the accessory.
4. Try to pair the accessory again."

Take Care!