I am unable to login to web player using my facebook account. However I am able to use the same account on desktop client.
Here is a screenshot of the error that i get.
Can you try logging in from an incognito window (chrome) or private tab (firefox). You'll need to type in your facebook credentials so it will be interesting to know if you get the same error.
The issue is fixed now. Something weird was going on with Spotify.
Enjoy your music :)