"As the supreme head of state for Vatican City, I would surmise that his status is no different to Mugabe etc."
Read above, yes it is, it's very different.
The Pope is the Head of the Holy See and it is the Holy See that he and his ambassadors represent and it is the Holy See that issues those Diplomatic Passports. Likewise it is the Holy See that countries have diplomatic relations with.
The Vatican City State was created in 1929 by Mussolini as a bit of hand waving to try and get around the whole issue of the Pope not being a proper "Head of State".
However the Church kept the separation of the territory of Vatican City and the Holy See. So the Holy See has no territory, which by international law means that it is not a state, but on the other hand it has recognised ambassadors (and had them even before the creation of Vatican City State), which only a state can have.
So his status is different and questionable.
I'm not saying that he isn't a Head of State, or that is is. What I am saying is that you can't claim that it's simple and obvious.