To me, "Jr" implies there was never anyone of the same name extant concurrently with the "III" except the father. The "II" leaves open the possibility there were predecessors to the father.
I know -- debatable, nitpicky point, from a former "II" who never used it. Actually, Major, Magnus, and Minor (like same-named boys in a Brit public school) would be classier. But why bother?