Actually, they’re the perfect example of the difference. From the days when they held Cyprus, or Malta, and St. Croix in the Virgin Islands, the Sovereign Military Order of At. John of Jerusalem (Hospitalers) of Malta has fallen on hard times. Thiough they still exist, they hold no sovereignty over any land anywhere. Their territories, a couple of villas in suburban Rome, are not their territory, but owe sovereignty to Italy, and belong to the Knights as landowners in fee simple and under their rule by the principle of extraterritoriality, much like the Cuban territory at Guantanamo belongs to the U.S. The underlying sovereignty is Italian on the one hand, Cuban on the other; the governing body holding it from Italy or Cuba is, respectively, the Knights and the U.S.A.
In contrast, the Holy See/State of Vatican City (they’re not synonymous, but it takes a good international lawyer to clearly define the difference) holds the thousand acres of Vatican City in full sovereignty. Granted that Italy could take it over with little effort, such an action would be an aggression different only in degree, not in kind, from Iraq’s 1991 invasion of Kuwait, or Nazi Germany’s 1939 overthrow of Cesky.
There is, however, a nitpicky correction needed, of the idea that Vatican City is a remant of the former, much more extensive Papal States. The 1860-70 reunion of Italy conquered totally the Papal States. Pius IX fled to Castle Gandolfo, living in self-proclaimed exile. To regularize relations with the Holy See, which still had extensive influence in Italy, Mussolini negotiated the Lateran Treaty, which gave the Pope sovereignty over the territory called Vatican City. Between 1870 and 1929, the Pope was in the same state as the Knights of Malta presently are: certain holdings informally granted extraterritoriality (Castel Gandolfo being a good example) but no land held in sovereignty.
In passing, Lichtenstein gained full independence in 1806 in something similar – prior to that date, they were a fief of the Holy Roman Empire, no different from Bremen, Oldenburg, or Saxe-Altenberg. But the point was that the House of Lichtenstein held from the Emperors, not from the Habsburg Archdukes of Austria, even though they were the same people. So when the H.R.E. was dissolved in 1806, its suzerainty over Lichtenstein, a legal fiction, vanished, and the Princes of Lichtenstein owed allegi8ance to – no one! Hence they were independent.