Spotify has a better free service
Apple's free service is extremely limited -- you only get radio stations. You won't be able to request and listen to individual songs, find and listen to playlists, or do much else you'd expect in a streaming music service. Spotify's free services varies according to whether you listen on a phone or via the desktop or tablet, but among its features in various platforms are listening to individual tracks on demand, finding and listening to playlists, sharing music and playlists, and more.
Spotify works on Android
Spotify works on Android. Apple Music doesn't. Apple says in the fall it will work with Android. But for now, Spotify does that, and Apple Music doesn't.
Spotify works on Sonos
Sonos is the best wireless music system available. I'm a big fan -- I've got half-a-dozen Sonos speakers throughout my house. Spotify works with Sonos. Apple Music doesn't. Apple Music is expected to work on Sonos systems by the end of the year. But if you want streaming music on Sonos now, you can't do it with Apple Music.
Spotify works with Facebook
You can connect Spotify to Facebook, and share your music with your Facebook friends. You can't do that with Apple Music.
Apple owns Apple Music
Apple has its tentacles everywhere, and along with a handful of other companies increasingly dominates the tech world. Choice is good; competition is good; variety is good. You vote for all of that by choosing Spotify over Apple Music.