What does the Nicene Creed mean when it says that the Father and the Son are of the same substance?

Nobody knows what the substance of God is. He exists outside our universe of time, space, and matter. Or rather, our universe exists somewhere in God. He gave the energy and wisdom that brought the universe into existence 13 billion years ago.

Angels are spirits, but they are still part of our universe. In that sense, their substance is still defined in terms of the “substances” of this universe. Perhaps their substances are energy waves or magnetic fields. Who knows? We will know one day. But my point is that the substance of God is completely different from even the spirit substance of the angels.

The Bible says nothing about the substance of God, except perhaps Hebrews 1:3, which says that the Son is "the exact representation of" God's substance. But it seems as if Arius and his supporters, in the time before Nicaea (325), argued that the Son is of a different substance than the Father for the creed condemns all people that say that “the Son of God is of a different hypostasis or substance” (than the Father). In other words, Arius and his crowd argued for a substantive difference between the Father and the Son (pun intended).

The creed responded that they are h*m*oúsios, meaning “same substance” (from the Greek words h*m*s, same, and ousia, essence). (Merriam-Webster)

After the Nicene Creed was formulated in 325, the church reacted strongly against the word h*m*oúsios and developed many creeds to replace the Nicene Creed and, in particular, the word h*m*oúsios. These views include the following:

DISSIMILAR SUBSTANCE - Heteroousianism held that God the Father and the Son were different in substance and/or attributes. These were called the extreme Arians. They described the Son as "unlike" (anomoios) the Father. (Encyclopedia Britannica, 1979, Arianism, Vol. I, p.509) For example, Emperor Constantius (Constantine's son) requested a council at Constantinople in 359 where the Heteroousians (dissimilar substance) defeated the Homoiousians (similar substance) in an initial debate. (See council of Arinimum.)

SIMILAR SUBSTANCE - Homoiousianism (from hómoios, "similar", as opposed to homós, "same, common") maintained that the Son was "like in substance." They were called Semi-Arians.

NO SUBSTANCE - Homoeanism declared that the Son was similar to God the Father, without reference to substance or essence. They held that the Father was like the Son in some sense but that even to speak of ousia was impertinent speculation. Hanson, RPC, The Search for the Christian Doctrine of God: The Arian Controversy 318-381 AD, lists twelve creeds that reflect the Homoian faith.

ONE SUBSTANCE - However, later in the fourth century, spearheaded by the Cappadocian fathers, the concept developed that the Son is not only of the “same substance” as the Father but also “one substance” with the Father. This later became the standard Trinitarian formulation. While all the other views held that the substances of the Father and Son are distinct, even though they may be the "same substance" (same type of substance), just as your and my substance is the same but distinct, the “one substance” view proposes that they ‘share” one single substance.

Note that the Nicene Creed of 325 is not Trinitarian in the sense of three Persons but one single substance. That concept was developed later. The emphasis of the Nicene Creed was merely the equality of the Father and the Son.

In Reformed circles, where I grew up, the “one substance” view is often expressed by saying that God is three Persons but one Being.

You ask: How are the Son and the Father different? You added: We cannot say that Jesus is God in the sense of identity. The different views above will give different answers. But the orthodox answer, as expressed by the Athanasian Creed, is that they differ only with respect to relations, namely:

• That the Father begets the Son and
• That the Holy Spirit proceeds from the Father and the Son (in Western orthodox theology).

But if they share one single substance and one single mind, as per the orthodox understanding of the Trinity, what is the practical difference between the three Persons? This is addressed in the question about Modalism.

To say "The Son and the Father are of the same substance" (consubstantialem as the word is in the Latin Niceno-Constantinopolitan Creed) is to say that they have the same being (ousia in Greek); that is, that they are the same kind of thing—God.

Traditionally, Catholic theology has said that even though the Son and the Father are of the same substance (consubstantiales in Latin; a semi-reasonable translation of the Greek h*m*ousioi), the Son proceeds from the Father. Thomas Aquinas uses the simile of a word or a thought, which is part of a person (in that thoughts are part of one's mind), but is also in a sense distinct from, and "comes from", one's mind.

Procession, therefore, ... is to be understood by way of an intelligible emanation, for example, of the intelligible word which proceeds from the speaker, yet remains in him. In that sense the Catholic Faith understands procession as existing in God. ... it is clear that the more a thing is understood, the more closely is the intellectual conception joined and united to the intelligent agent; since the intellect by the very act of understanding is made one with the object understood. Thus, as the divine intelligence is the very supreme perfection of God, the divine Word is of necessity perfectly one with the source whence He proceeds, without any kind of diversity.

(Summa Theologica, First Part, Question 27, Article 1)

The term the Nicene Creed uses for substance is h*m*oúsios. This term was intentionally chosen to separate the Creed from various forms of Arianism that denied the divinity of Jesus. The Nicene Creed is arguing that Jesus is fully divine just like the Father. The common forms of nontrinitarianism at the time commonly denied the divinity of Christ by arguing that he was similar to the Father in attributes, similar in substance to the Father, or different in both attributes and substance. The Nicene Creed explicitly states the 'one in substance' to both affirm the divinity of Christ as well as affirm the truine Godhead. When communicating it in contemporary English, h*m*oúsios is sometimes translated as 'of the same being.'

It is the foundational conception of the Trinity of all major forms of Christianity still in existence: Catholic, Orthodox, or Protestant. So no, it is not a distinctly Catholic belief. The substance in question is the 'divine being' of the Godhead: Father, Son, and Holy Spirit. Each is of the same substance in that they are all fully divine, inseparable yet distinct. It is through this concept of substance that Christians describe the Trinity as 'one God in three persons.' The three persons of the Trinity all share the same divine 'substance' in the sense that they constitute one God -- but they are each distinct from each other in their persons.