I read an article that has said there has not been an islamic caliphate for 90 years since the end of the first World War (Ottoman Empire).
I was under the impression that both Iran and Saudi Arabia were under a caliphate.
If this is not the case, would someone mind explaining to me the difference between a caliphate and the islamic rule of these states?
The caliphate represents a centralized seat of Islam. The Caliph doesn't necessarily administer the governmental responsibilities of a state, though they may be head of state as well, but they have broader ceremonial authority and responsibility over the whole Islamic world, whether they're under their state or not. So, for example, the Ottoman Sultan would claim spiritual authority over all Muslims, even ones in India and Iran which were outside his empire.
It's somewhat similar to the papacy and how the pope is both nominal ruler of Vatican City and also the spiritual leader of all Catholics.