An administrator with the will annexed is appointed when a person dies with a Will, but the Will doesn't name an executor (or when the executor(s) named in the Will can't or don't want to act). For instance, my Will names my father as primary executor and no successor executor.  My father dies, and then I die.  I have a valid Will, with provisions that relate to the disposition of my property, but no one is able to carry out (or execute) the terms of my Will.

An administrator de bonis non is appointed if an acting administrator or executor ceases to act (perhaps because of death or disability), and no one is available to take the administrator's place.  "De bonis non administratis" means "of goods not administered," according to Wikipedia.