What is the difference between "person" and "substance"? According to the answer here:


  St. Thomas Aquinas explained the preceding definition [of person] in terms that practically constitute a new definition: a substance, complete, subsisting per se, existing apart from others (Summa Theologica, III, Q. xvi, a. 12, ad 2um).


According to trinitarian theology, the triune god is said to be "three seperate persons" consisting of "one substance". 

Does this mean the triune god is "three separate substances, each existing apart from other substances, consisting of one substance"?

In other words, what are the key differences between a "person" and a "substance"?
The terms substance and person are (in the context of Trinitarian theology and Christology) Latin translations of the Greek terms ousia and hypostasis.

The background from Greek philosophy.

Plato and Aristotle were the ones who first formulated the terms ousia (substance or essence) and hypostasis (individual substance) in a philosophical sense. Plato did not make any treatises; it was his student Aristotle who was the first to deal with substance (ousia) in a systematic way.

Substance

A lot could be said, but, briefly, a substance is any individual being that exists separately from—and is not merely a characteristic “inherent” in—other beings. For example, an oak tree is a substance; its color, size, weight, location, and so forth, are not substances but accidents. The tree exists “separately” from other substances; the tree’s accidents cannot possibly be separated from it.

As Aristotle famously remarked,


  “Being” (τὸ ὂν) is used in many ways, but referring to one, and one nature only, not ambiguously (Metaphyiscs 1003α32; my translation).


The “one thing” to which being always refers is οὐσία; that is, substance (1003β5). In other words, although the verb “to be” can be used in a lot of different ways, it always refers ultimately to some substance. For example, we say, “The tree is green,” or “The tree is in such-and-such a place,” and so forth, but we can affirm neither greenness nor location without a tree (the substance) to go with it.

Note that for Aristotle (in opposition to his teacher Plato), substance primarily denotes individuals, not universals. That is, for Aristotle, an individual tree is a substance in the primary sense; tree-kind can also be called “substance,” but only in a secondary sense.

In contrast, Plato thought that reality is to found principally in the ideas: hence, for Plato tree-kind is more real than individual trees. (See, for instance Plato’s Phaedo 78c–79a, and 100b–101d.) Not so for Aristotle.

This distinction becomes important when it is applied to Trinitarian theology and Christology, because those who formulated that theology were fundamentally Platonists, not Aristotelians.

Hypostasis and Person

The term hypostasis simply refers to substance in (what for Aristotle is) its primary sense: an individual. The Greek term can refer to any individual with a separate existence: a tree, a human being, a dog, a stone, and so forth.

The Latin term person—according to Boethius’ famous definition—refers to an “individual substance (hence a hypostasis) with a rational nature” (Boethius, Liber de persona et duabus naturis, chapter III, PL 63:1343C). In other words, the term person in Latin has a more restricted meaning than the Greek hypostasis—that is, it is limited to substances capable of intellectual knowledge (men, angels, and God). In a Trinitarian or Christological context, however, the two terms are identical in meaning.

Substance and Hypostasis (Person) in Trinitarian theology

When the Council of Nicaea condemned the heresy of Arius, it affirmed that the Son is one in substance (homousios) with the Father. (See text of the Council of Nicaea in Denzinger-Hünnermann [DH], Enchiridion symbolorum, 125.) The council fathers, however, failed to distinguish well between hypostasis and ousia, as is evidenced by the following condemnation:


  Those who hold that the Son of God is of a different hypostasis or substance [ousia] from God … are anathematized by the Catholic Church (Council of Nicaea, DH 126).


This condemnation is not false, but was written at a time when the terms hypostasis and ousia were not yet used in the technical sense they have today. (The Nicene condemnation reflects the fact that in most of the philosophical discourse of the day ousia and hypostasis were essentially synonyms.)

It was the Cappadocian Father Basil of Caesarea who first made that distinction (probably influenced in part by Origen). It should be noted that Basil’s training was Platonic, not Aristotelian, so he was drawing from the Platonic interpretation of the terms substance (ousia) and hypostasis. Basil, therefore, interprets ousia to mean universal or common nature (e.g, tree-kind, as opposed to individual trees) and hypostasis to mean a concrete individual.

It is, however, very important to note that Basil went to great lengths to explain that in applying these terms, he was not attributing them to God in the same way that they apply to creatures. Among creatures, individuals belong to “genera” or “species” that are constituted by other individuals of the same kind. Thus, for instance, oak trees belong to a species of other trees that are also oak trees; human beings belong to the species called “man,” and so forth.

The Hypostases (Persons) of the Trinity, however, are not “individuals” in the “species” called God; such would be tantamount to tri-theism. Rather, each Hypostasis possesses the Divine Substance (and the very same Divine Substance) in its entirety. (That is possible, as Gregory Nazianzen explains, because each Hypostasis is fundamentally a relation of origin with respect to the other Hypostases [Orat. 31 de Spiritu Sancto, IX, PG 36:141C].)

The Persons (or Hypostases) of the Trinity, therefore, are immanent relations of origin that operate within the Godhead. They are the subjects of God’s action and answer the question Who?

The Divine Substance, on the other hand, answers the question of what kind of being God is.

Again, it should be stressed that the Persons do not “divide” the Divine Substance in any way; God remains utterly simple and undivided.

Substance and Person in Christology

The terms ousia (and its synonym physis or “nature”) and hypostasis also appear in the context of Christology.

The vast majority of Christians believe that Jesus is fully God and fully man; the technical way that Chalcedonian Christians affirm this is to say that Jesus has two natures (physes). Evidently, Jesus’ Divine Nature (or Substance) is the same one that he has always had as the Divine Son; in this regard he is one in substance (homoousios) with the Father. At the Incarnation, he acquired a human nature; in this regard, he is one in substance (homoousios) with man. (See the definition of the Council of Chalcedon, DH 301-303; note that that the Council of Chalcedon, in affirming this, understands ousia in the Platonic sense—even though, of course, it recognizes that the Divine Substance is utterly one had does not have “individual members” the way the human species does.)

However, Jesus is one hypostasis or Person, not two, as was clarified in the Council of Ephesus. (See DH 250-258.) Over the course of the century that followed the Council of Chalcedon (451), it gradually became clearer that the Hypostasis (Person) of Jesus is the very same Hypostasis that is the Son, that is the Second Person of the Trinity:


  there is but one hypostasis [or person], which is our Lord Jesus Christ, one of the Trinity (Council of Constantinople II of 553, DH 424).


In Christology, just as in Trinitarian theology, “nature” or “substance” answers the question, “What kind of being is Jesus Christ?” and the answer, is “God and man.” 

“Person” or “hypostasis” answers the question, “Who is Jesus Christ?” and the answer is, “the Divine Son, the Word, the Second Person of the Trinity.”

In answer to the O.P.’s specific questions:

Is the triune God “three separate Persons”?

No, the Persons are in no way separate from each other. The are really distinct (that is, the Father is not the Son, and the Son is not the Spirit, and so forth). However, they cannot be considered “separate” from one antother, because each person is perfectly identical to the Divine Substance. (Each Person possesses the Substance in its “entirety”—which we say very improperly, because God does not have parts—and possesses the very same, undivided Divine Substance as the others.)

Of course, God is infinitely “separate” (in the sense of infinitely independent) from His creatures—which is what makes Him, so to speak, the Substance par excellence. (He is not, however, “separate” in the sense of “distant;” indeed, there is no one more “present” to creation than He is.)

Does this mean the triune God is ”three separate substances”, each existing apart from other substances, consisting of one substance?

No. God is a unique, undivided Substance. He is unique, undivided, and also triune; that is, within the Godhead, there are three subjects that can be the answer to the question “Who?” The Persons (or Hypostases) differ from one another only in relation of origin (Fatherhood, Sonship, and Procession); in every other respect—that is, with respect to the entire Divine Substance—they are perfectly identical.