The Holy See is not a state. Its the Pope’s realm over the Church. The Holy See is all over the world. The Vatican is a state. Its realm is geographic. The Vatican is one of very few states that is not a member of the United Nations. They do send a priest to listen in on the UN and walk around the halls.
This is pretty confusing.
Consulting the CIA World Fact Book (LINK), you will read that: "The United States and the Holy See announced the establishment of diplomatic relations in 1984."
"The Holy See and the United States both are members of the Organization for Security and Cooperation in Europe. The Holy See also is an observer to a number of international organizations of which the United States is a member, including the United Nations, Organization of American States, and World Trade Organization."
"The U.S. Ambassador to the Holy See is Kenneth F. Hackett; other principal embassy officials are listed in the Department's Key Officers List."
It seems "The Holy See" is the name of the government of the Vatican City.
Yes, yes, I know that "See" comes from "Sede" which means "Seat or Chair," and it refers to the chair of the Bishop of Rome, that is, the Pope, considered as spiritual leader.
So it seems to me that the "secular government" ought to be called "Vatican City" and the purely ecclesial organization ought to be called "Holy See" --- but that is not quite the case.
I'm somewhat cross-eyes from reading through all this stuff. I'm sure there are historic reasons and blah-blah-=blah.