If anyone is still having trouble with this... On windows 8, I right clicked the file after downloading and chose properties. I then clicked 'unblock' and 'apply'. That allowed me to install spotify without getting the error that a non-administrator account is required.
just right click on the installer, and click properties. Then click "compatibility" and check the box that says "run in compatibility mode" and make sure the dropdown is selecting windows 7. Click OK, then run. Voila.
Thanks for the tip but I also found that if I open the file from my downloads folder in file explorer it installs ok. It only causes problems if I try to install from the downloads icon in the browser. No idea why this is so.
I tried all these ways listed up here but nothing worked.. What worked for me was to not install withe the Spotify-Installer but to save the programm in a folder on my PC (it was named "FullSpotifySetup32") and when I double clicked on this, Spotify finally opened and worked again.