Interesting. I too have been dealing with this issue. Check your internet connectino icon, either wireless or LAN. does it have a yellow ! mark claiming no internet access? I disabled my AV software and went to services.msc, found "Network List Service" and restarted. The yellow ! went away and Spotify works again. I'm using Kaspersky AV, btw. This is probably a temporary solution for the time being, you'll have to repeat at restart if the false positive "no internet connection" alert shows up.
Re: Songs won't play in desktop app, but do play in web player