To completely stop the activity of Spotify in our timeline, go to Facebook and open 'Privacy Settings' from the drop down menu in the upper right corner.
Come to the "Applications and Websites" section and click on 'Edit Settings'.
Find Spotify in the list of applications you use and click its icon.
In the "Who can see posts and activity from this app on Facebook?" changes 'Friends' with 'just' and thus will delete Spotify from your Facebook Timeline instantly.