Anthophyta literally means "flower (antho-) phyta (plant)" which gives a primary characteristic of the group. Magnoliophyta takes its name from one of the more primitive members of the group - it was the convention that a taxon takes the name of the most important member of the next lower taxon, so the Magnolia (genus) as the most primitive flowering plant, was chosen as the root name for the family (Magnoliaceae), order (Magnoliales), class (Magnoliopsida), and division (Magnoliophyta). The system which uses Anthophyta defies this convention. The classes under Anthophyta would be Monocotyledoneae and Dicotyledoneae. This system might be more "politically correct" as it doesn't assign more importance to any single member of the group.
So the biggest difference is which system of taxonomy you're using, and I still see both names in use. But the trend seems to be toward moving to Anthophyta.