UPDATE based on latest documentation:
Android has included in its documentation on when you should use Service vs Thread. Here is what it says:
If you need to perform work outside your main thread, but only while
the user is interacting with your application, then you should
probably instead create a new thread and not a service. For example,
if you want to play some music, but only while your activity is
running, you might create a thread in onCreate(), start running it in
onStart(), then stop it in onStop(). Also consider using AsyncTask or
HandlerThread, instead of the traditional Thread class. See the
Processes and Threading document for more information about threads.
Remember that if you do use a service, it still runs in your
application's main thread by default, so you should still create a new
thread within the service if it performs intensive or blocking
operations.
Another notable difference between these two approaches is that Thread will sleep if your device sleeps. Whereas, Service can perform operation even if the device goes to sleep. Let's take for example playing music using both approaches.
Thread Approach: the music will only play if your app is active or screen display is on.
Service Approach: the music can still play even if you minimized your app or screen is off.
Note: Starting API Level 23, you should Test your app with Doze.
Android Documentation - Services