The main things that come to mind for me are:
1. Sometimes Spotify will refuse to play certain songs & won't give any indication of why. Closing and reopening the app seem to fix it. Or sometimes switching to a different device fixes it.
2. My listening history isn't carried across devices for some reason.
3. The contents of "daily mixes" and other generated playlists seem to change depending on the device being used.
It's not quite enough to make me use a different service, but it's annoying not being able to just pick up my phone and continue where I was on my computer