No need to logout / login each time on the Tivo App (It's a huge PIA to login using the remove).
Like all other Spotify apps, music will only play on one of your devices at a time. If you start playing music through the Tivo Spotify app, it will stop any active playback on your other devices. For instance, if you start playing music on your phone, the Tivo app will stop playback.
The Tivo Spotify App even allows you to setup multiple logins (We have a family account)...so once you setup your account there is no need to redo the entire login (unless you change your password).
Having said all that the Tivo Spotify app can use a lot of work. It's nearly impossible to use on its own. Setup playlists on your desktop or other devices and use playback on Tivo.
The Tivo Spotify app does not automatically advance to the next song in your playlist. Keep the remote at hand and constantly forward to the next song in your playlist. Sometimes it will advance...very mysterious.
I'm not sure who supports the app; Spotify or Tivo. I wish they would open source it.