Try to remove your local files and put them into your playlist again. Then connect your phone to your computer through a cable. Make sure your phone and computer is on the same Wifi and download the playlist again.
Check Windows firewall or if you have a anti-virus suite running that includeds a firewall, make sure to allow spotify.exe through. Under Windows Firewall there are two spotify.exe apps listed, one is checked for public networks and one for private networks. Put a check mark next to them both and click OK at the bottom.