Sorry for late answer, hopefully it sorted it out for You already, just in case if not:
in Vox Settings, check Your Spotify account - if it's not logged in, then just repeat login procedure again. Once logged in, synchronization begins. It will load the list of all Your "My Music" and "Playlists" so may take some time. This process is indicated by "Your Library" menu in Spotify section of Settings, reading, "syncing...". Once this field changes to orange "Sync" button, Your Playlists should be in-place in VOX Collections.
Welcome back should You have further questions and proposals!
This is how it should work, but it doesn't. I understand whole process, but in my case syncing just begins, then my Spotify playlists appears in the VOX (they are empty right now), but when it shows orange "Sync" info, almost all playlist are still empty - no songs, I can see just playlist name. Only very few of them (it's about eight, but I have a way more) are actually synced, exactly how I described it in my original post. There is an interesting thing I discovered: that few synced playlists are always exactly the same, no matter how many times I try to repeat syncing process or even delete VOX app and try it again, it always sync exactly the same playlists from my Spotify account and nothing more. Strange.
So unfortunatelly I can't use VOX with my Spotify account, it doesn't work for me and I don't know why.