If you're seeing playlists or saved tracks on one device but not on another, first make sure you're logged in on the same account on both devices.
If you are, try logging out and back in on the device that hasn't synced to shake things up. Make sure your apps are not in Offline Mode.
If that doesn't do the trick, make sure to remove all devices listed here and try a reinstall of the app on that device by following the steps here.
This Spotify Answer was submitted by @Nico.
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