With the mobile version of Google All Access, which is rolled into the Google Play app, what you need to know is that a little inconspicuous pushpin icon at the top of your playlists is how you sync tracks for offline listening. Just give the pushpin a tap and the playlist downloads to your Android phone (Google doesn't support iOS devices yet). By contrast Spotify's mobile app puts a conspicuous Download button at the top of playlists.
Streaming Radio: Google Win
One of my bad habits with any music service is playlist fatigue. I'm often too lazy to create new playlists, so I listen to the same ones repeatedly. That's why most music services offer "radio stations" that create playlists based on a genres or artists.
Between Google and Spotify, it's Google Play Music All Access that is the best. With Google's radio feature (available on the desktop and mobile) you can preview a list of upcoming tracks and delete or change the order of songs. Spotify's radio offering no such flexibility.
Music Discovery: Google Win
One of the best features with Google's All Access is something I never knew I wanted - music discovery. Thanks to Google's radio feature, mentioned above, I can click on an artist I like and a playlist is generated. Because I can peruse the playlist I can see what Google's thinks is related music. So far I've been impresses with Google's ability to find interconnected music and introduce me to new bands.
I think Spotify does a good job of generating radio playlists of related music. But Google's playlist preview feature is superior and has brought a lot of new music into my life in the short time I've been a Google All Access user.
Social Sharing: Spotify Win
Linking a streaming music services with a social network has its advantages and disadvantages. For those, like myself, it's a great way to share music with friends. Others find it intrusive. Neither Google or Spotify force you to merge your social life with your account, but you can.
Spotify users can share their playlists and what they are listening to with each other. You can even link a Spotify account with Facebook. If you so choose, you can have Spotify post to your Facebook news feed the music you're listening to.
(Image shows what sharing from Google music service to a Google+ profile page looks like.)