One thing I have that MIGHT be causing problem, but shouldn't about:config media.autoplay.enabled is set to FALSE
The reason this is set to this is to stop hundreds of websites that autoplay videos when you open their page (CNN is a big one that does it, but many others) All this flag does it make it so you have to click on play button first before media content will play. So all that should happen is when I click on a song, it should still play if I hit the play button. But Spotify is not regonizing the play button as activation.
In Firefox, if the about:config flag "media.autoplay.enabled" is set to false then the spotify webplayer will NEVER work. This flag is important to stop nasty web sites that bombard users with videos or ad-videos the user has no control over. Please allow the web player to work correctly with this flag set to false. I would have no problem with the music playing once I click play, but this flag stops a-hole sites that load a video without waiting for user to click play.
Type in URL bar: about:config What is your "media.autoplay.enabled" flag? I'm betting it's set to true. However, due to so many **bleep** websites adding auto play videos and autoplay ads that blast the user without their consent, most people change that flag to False. The way it's SUPPOSED to work, is the user has to hit the PLAY button videos etc. for the media to play. For some reason, the spotify webplayer is not accepting the play button as activation. WIth this flag at true, the player works. But, myself and many others refuse to have this flag on true because of stated above reasons. Chrome has a similiar flag, (they call it "user gesture) but because of their ad network and revenue, it doesn't work on most sites anyway because crappy sites found a loophole around Chromes implemention. I refuse to go back to using a web that Chrome decides for me.
I'm running Ubuntu 16.04, Firefox 57.0.1 64 bit. Encountered the same issue. Steps I took to fix this issue: 1. enable widevine. 2. install VLC media player. 3. run sudo apt-get update. VLC seems to install some necessary dependencies for open.spotify to run properly.