"As the supreme head of state for Vatican City, I would surmise that his status is no different to Mugabe etc."
This gets a bit complicated....
The Pope is the head of the Holy See and ambassadors from the Church are ambassadors for the Holy See.
Vatican City is the current home of the Holy See and has been since Mussolini and 1929 but it is not the Holy See. The Holy See has nearly moved it's physical home several times throughout history.
A state is defined under international law as having a defined territory. The Holy See doesn't, but it does have recognised ambassadors, which it what gives it it's status despite the lack of territory.
So its confusing and very different from Mugabe.
I'm not saying the Pope isn't a "Head of State", but what I am saying is that's it's complicated and far from clear cut.
The previous visit of a Pope back in the 1980s wasn't seen as a visit by a Head of State, so this is something different.