The actuality operator is usually not interpreted as a quantifier, it indicates that what follows belongs to the privileged word, the actual world. The existential quantifier, on the other hand, quantifies, and over things in whatever possible world. So ☐(∃x P(x)) for example says that there exist objects with property P in every possible world, i.e. they exist necessarily. In some theories ∃x is even  used unrestrictedly, i.e. quantifies over all possible worlds, as in [Lewis's counterpart theory][1]. Then one can combine the two as in ∃x Ax, which says that the actual world has things. In some non-modal theories there is a similar distinction between the [existence predicate][3] and the existential quantifier, when people (Meinongians) want to quantify over non-existent things but without committing themselves to their existence. So ∃x(¬Ex) says that some things do not exists like ∃x(¬Ax) says that some things are not actual.

However, [Hazen in Actuality and Quantification][2] does introduce actuality quantifiers:"*Ordinary, world-restricted,  quantifiers  are interpreted  as ranging over  existents; their logic is formalized  by putting
 existence premisses/hypotheses  into the familiar  rules. Actuality  quantifiers  are interpreted as ranging  over  things that actually exist
 (actual existents)*". This is to satisfy [actualists][4], who would only allow quantification over actual objects, not modal abstractions.


[1]: http://www.ulrichmeyer.org/wp-content/uploads/2012/09/Counterparts_Actuality.pdf
[2]: http://projecteuclid.org/euclid.ndjfl/1093635586
[3]: http://plato.stanford.edu/entries/actualism/
[4]: http://plato.stanford.edu/entries/actualism/