The Life Training Institute wrote a post on this unusual situation, quoting from Maureen Condic's article, "A Biological Definition of the Human Embryo." The bottom line:
If the necessary structures (molecules, genes etc.) required for development (i.e., an organismal level of organization) do not exist in an entity from the beginning, the entity is intrinsically incapable of being an organism and is therefore not a human being.When we say "human beings begin at fertilization" we are actually using a sort of shortcut. It takes a lot longer to say "human beings are human organisms that have begun the course of human development." In the vast majority of cases, fertilization is the point when a new human organism begins the course of human development, and so we just say "fertilization." However, there are some fertilized eggs that are not organisms (hydatidiform moles), and there are some organisms that do not begin human development until well after fertilization (the second embryo in a pair of identical twins).
If fertilization is usually--but not always--the beginning of a new human being, does this mean we don't really know when human beings begin? Nope. We know. I will say again: human beings begin when a new human organism begins the process of human development. This means the second embryo in a pair of twins is a human being, and a hydatidiform mole is not a human being.
Check out the rest of The Life Institute's post for some great analogies between hydatidiform moles, the Alphabet Song, and wildlife spotting. :-)