yeah,it seems that my chrome book has this issue from time to time too,a friend tought my to clear your history on chrome,sometimes spotify updates and for some reason chrome takes you to the old version so it doesnt process it.
It seems to work for me now. What happened? I was closing a program in task manager when in the corner of my eye i saw another program appear there. It was SpotifyWebHelper.exe. I had a feeling that i should try the web player again and it worked.
I did a search and indeed this program's purpose seems to be to allow users to open up and play Spotify's music in the web player. Why it wasn't running before, i have no idea.
edit: I spoke too soon. At first it worked, i could use the web player at work without the client. Now it doesn't work anymore and my work pc has the same problem as before. It doesn't make a difference if i have the client open on my home pc or not. I give up.
I've been having this problem in Explorer, but I just cleared cookies and data, but also, in Explorer's Internet Options dropdown, selected the Security tab and added spotify.com to trusted sites. Seems to be working now....