I upvoted @Doc's answer, and then found this link:
What are Europe's two closest capital cities?
Essentially it's going to come down to your definition of a city (and you thought defining a country was hard!).
Their definition is that the two closest capitals in Europe are Rome and the Vatican - and at a distance of 0km, make them the closest in the world.
However, they explain by pointing out that the Vatican City is itself a sovereign state as well as a city, and therefore not a true capital of a country, making Bratislava and Vienna the closest in Europe (for the purpose of their question, however it would make Brazzaville and Kinshasha the closest in the world for the purpose of YOUR question)
However if you're asking the two closest capital cities then as the state 'Vatican City' comprises of more territory than just the city 'Vatican City', then yes, you could argue that Vatican City and Rome are the closest in Europe and the world.
So really, it comes down to your definition, and if you count Vatican City as a capital city, then yes, that's the closest (to Rome), if not, then it's Brazzaville and Kinshasha.