Summary:

If you are using a Samsung device which has been updated (or running) Android 6.x Marshmallow it appears that they have introduced an agressive power saving option which is closing Spotify or stopping it accessing data when the application is sent to the background or the screen display turned off. The following are common symptoms caused by this:

Symptoms:

• Next song in an album/playlist is not played
• Lock Screen control vanishing
• Spotify notification disappears
• Tracks marked for offline listening fail to download
• Notifications for new followers, albums etc have not been coming up

Resolution:

If you are using a Samsung device and experiencing any of these symptoms please disable the Power Saver settings as follows: 

Settings > Battery > Power Saving Mode> In the Power Saving Mode settings disable the "Restrict background data" setting

Kudos to @Cerda45 for the tip here

If it's still not working after doing the above then disable Power Saving Mode entirely:

Settings > Battery > Power Saving Mode

Settings > Battery > Detail > Spotify >Turned off

Further Info:

I believe Samsung are aware of the issues this is causing - not only to Spotify - and we can hope they improve the power saving feature in the next software update which is hinted at being sometime around June.

Still having the issue? This could be the Android 'Doze' feature taking effect. Read the section "Tracks stop playing when screen turned off or the Spotify application is in the background" in the [COMPLETE GUIDE] How to fix most Android issues