I have experienced this problem a few times with my parent's iPhone.  It is frustrating when you don't know the answer, and even when I've searched and tried the community answers.

It may help to put local files in their own play list for ease of organisation.  Mark this list as Available Offline.

Next; both the the phone and computer need to be on the same Wi-Fi network.  This sould be done through a Wi-Fi router.  Ad Hoc networks do not work for this function.

The phone should appear under devices in Spotify.  If it does not, it means you are not connected.  You need to re input the Wi-Fi password on the phone as if doing it for the first time.  Check again to see the phone appears under devices in Spotify.  You can select from there which lists you want to synchronise.