What is the difference between setDisplayHomeAsUpEnabled and setHomeButtonEnabled?

For what you want to do, actionBar.setDisplayHomeAsUpEnabled(true) is enough.

For the difference :
actionBar.setHomeButtonEnabled(true) will just make the icon clickable, with the color at the background of the icon as a feedback of the click.
actionBar.setDisplayHomeAsUpEnabled(true) will make the icon clickable and add the < at the left of the icon.

As Android says:

- setDisplayShowHomeEnabled(boolean showHome)
  // Set whether to include the application home affordance in the action bar.
  // (and put a back mark at icon in ActionBar for "up" navigation)

 -setHomeButtonEnabled(boolean enabled)
  // Enable or disable the "home" button in the corner of the action bar.
  // (clickable or not)

It should be quite clear i think