Can someone tell me the TRUE difference?
Use Service when you've got something that has to be running in the background for extended periods of time. It's not bound to any activity. The canonical example is a music player.
AsyncTask is great when some stuff has to be done in background while in the current activity. E.g. downloading, searching for text inside a file, etc.
Personally I use Handlers only to post changes to the UI thread. E.g. you do some computations in a background thread and post the result via handler.
The bottom line: in most cases, AsyncTask is what you need.