It seems to me that the formulas:
are quite similar, if we translate the modal propositional ones into PWS. Here is the result:
Translated into a formal-ish english language, they should be read as:
1*. There exists a possible world, w, such that P is the case in w.
2. There exists an x such that Fx.
3*. For all possible worlds, w, P is the case in w.
4. For all x, Fx.
The similarity becomes stronger when an interpretation is added to the predicate logic formula though I leave that up to the reader to do.