DialogFragment advantages over AlertDialog

This is easy.
DialogFragment is a fragment. So what can a fragment provide you while other objects can't?
It's the lifecycle callbacks.
So with DialogFragment, it can be very powerful and makes your code much cleaner.
Have you ever seen window leaks if you didn't close a dialog when its Activity was getting destroyed? So to prevent that, have you ever tried to close the dialog when onPause() was called? So to do that, have you ever had to make a reference of that dialog to a class level object?
With DialogFragment, it's all handled.
And you get all lifecycle callbacks.
Then you can provide more intelligence to the dialog and make it do some smart work on its own rather than Activity telling it what to do.

Use DialogFragment over Dialog:

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• Since the introduction of API level 13:

  the showDialog method from Activity is deprecated. Invoking a dialog elsewhere in code is not advisable since you will have to manage the dialog yourself (e.g. orientation change). Not using the showDialog will result in occasional exceptions, the dialog is not linked to any Activity.

  Note about showDialog:

  reference of Dialog: Activities provide a facility to manage the creation, saving and restoring of dialogs. See onCreateDialog(int), onPrepareDialog(int, Dialog), showDialog(int), and dismissDialog(int). If these methods are used, getOwnerActivity() will return the Activity that managed this dialog.

• Difference between DialogFragment and AlertDialog

  One thing that comes to mind when reading your question. Are they so much different? A DialogFragment is pretty similar to a Dialog, it's just wrapped inside a fragment. From Android reference regarding DialogFragment:

  A DialogFragment is a fragment that displays a dialog window, floating on top of its activity's window. This fragment contains a Dialog object, which it displays as appropriate based on the fragment's state. Control of the dialog (deciding when to show, hide, dismiss it) should be done through the API here, not with direct calls on the dialog.

• Other notes

  • Fragments are a natural evolution in the Android framework due to the diversity of devices with different screen sizes.
  • DialogFragments and Fragments are made available in the support library which makes the class usable in all current used versions of Android.