Spotify is using a streaming installer in order to get the necessary files. The main causes for installation error are directly related to admin rights, firewall, and network issues.
Disable the firewall in your computer for the time being, and then download the package from the URL listed below. Right click it and choose Run as Administrator.
You can disable the firewall using Start > Control Panel > System and Security > Windows Firewall > Turn off Windows Firewall (left section)
Download URL: https://download.spotify.com/SpotifySetup.exe