What is the difference between the Vatican and Vatican City, and how do their roles and functions differ within the context of the Catholic Church and international relations?

The Vatican is the headquarters of the Roman Catholic Church, led by the Pope. Vatican City is a sovereign city-state within Rome, where the Vatican is located. The Vatican is the spiritual center of the Catholic Church, while Vatican City is a political entity with its own government. The Vatican's role is primarily religious and spiritual, while Vatican City's role is more focused on governance and diplomacy in international relations.