I have been using Pandora and iTunes Radio, and have heard about Spotify. Apple's soon-to-be completed purchase of Beats Music intensified my interest in understanding and communicating the difference.
Pandora and iTunes Radio enable users to create the equivalent of a radio station based on a song or artist. With both, listen to ads or pay to listen ad-free. This means that a playlist is made for users.
Spotify enables users to play specific desired songs at will, in playlists. The users make the playlists.
Beats music is like a modification of Spotify, with playlists to let users explore and get familiar with musicians that are all built by humans.
I buy most songs I want, and make my own playlists. However, services are great ways to explore music.
Time magazine has a decent article describing the features of many online music services.