How do “metaphysics” and “ontology” differ?

In 2001, I asked Merriam-Webster the following question. Emily Arsenault from the editorial department replied as follows.

Q: How do “metaphysics” and “ontology” differ?

A: Their entries from Collegiate Encyclopedia might help:

ontology: Theory of being as such. Ontology is synonymous with metaphysics as defined by Aristotle, but because metaphysics came to include other studies (including philosophical cosmology and rational psychology), ontology has become the preferred term for the study of being. In the eighteenth century, C. Wolff understood ontology as a deductive discipline leading to necessary truths about the essence of things. Wolff’s successor, I. Kant, presented influential refutations of ontology as a purported system of knowledge. Ontology against became important in the twentieth century, notably among students of phenomenology and existentialism, particularly M. Heidegger.

metaphysics: Branch of philosophy whose object is to determine the real nature of things, to determine the meaning, structure, and principles of whatever is insofar as it is. In the history of Western philosophy, metaphysics has been understood as:

1. an inquiry into what sorts or basic kinds of things (e.g., the mental and physical) exist
2. the science of reality as opposed to appearance
3. the study of the world as a whole
4. a theory of first principles

The term means literally “what comes after physics,” and was used to refer to the contents of Aristotle’s treatise on what he himself called “first philosophy.” Aristotle had distinguished two tasks for the philosopher: to investigate the nature of the properties of what exists in the natural, or sensible, world, and to explore the characteristics of “being as such’ and inquire into the character of “the substance that is free from movement,” the unmoved mover. The first constituted “second philosophy” and was carried out primarily in his Physics; the second, which he also called “theology” (because God was the unmoved mover), is discussed in his Metaphysics.