Aristotle wrote that being is said in many ways. Though he did twice mention a possible “science” of being qua being, in both of the books of the Metaphysics in which he starts to discuss it, anticlimactically the only content he gives it is the principle of noncontradiction.
Metaphysics was a title assigned by a later editor to a collection of manuscripts of different dates. Commentators debated about its true subject matter. The idea that metaphysics equals ontology — opposed in the middle ages by the highly influential Averroes — became dominant only relatively late. The equation with ontology was especially associated with projects significantly different from Aristotle’s, like those of Avicenna and Duns Scotus. Eventually it became canonical with Wolff.
Heidegger wanted to distinguish Being from beings, and spoke about a forgetting of Being after the pre-Socratics, who allegedly had it in view. I say good riddance, if there ever was such a thing.
It is true and good that there is no Being in Aristotle. He correctly said “being” is not a unitary concept. The core of the Metaphysics is instead about what he calls ousia, or what answers the question “what a thing was to have been” — i.e, form, not being as existence in the common modern sense. He also mentions other sorts of being, such as being the case or being true.
Kant correctly pointed out that existence is not a property. Hegel in the Logic correctly said Being is empty and equivalent to Nothing.
In Greek, “existence” literally means standing out. To “exist” in this sense is to be determinately distinguishable or, in modern terms, to be a subject of some existential quantification, as when we formulate a mathematical proof that for any given A with specified properties, there “exists” (i.e., we can pick out) some B or a unique B with specified other properties. Existence in the sense of standing out is always relative to something else.
An abstract, nonrelative concept of “existence” is not needed in order to express real-world constraints and determinacy. Aristotle for instance uses more specific and supple concepts for this, like energeia (“at-work-ness” or actuality) and dynamis (potentiality).
The notion of existence as a nonrelative property of a thing, I suspect, owes something to the concern of medieval theologians to prove such a property for a nonobservable entity.
If something is normatively important, it is so regardless of whether it “exists” or is ideal or virtual. What is practically important is not abstract existence but practical difference, normative importance and conceptual articulation. (See also Form; Aristotelian Dialectic; Johnston’s Pippin.)