Sim, você pode. A FAQ diz o seguinte :
You can install Spotify on as many computers as you like. You may be logged in to Spotify from several computers simultaneously, but you can only play music on one computer at a time. The music at your other computers will automatically be paused. If you want to install Spotify on more than one computer simply download the program again and log in with your existing username and password, there is no need for an invite token.
As suas listas de reprodução estão associadas à sua conta do Spotify, pelo que serão as mesmas em todos os computadores onde instalar o Spotify.