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 Px) 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. 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 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 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, who would only allow quantification over actual objects, not modal abstractions.