Objection 1. It would seem that in God the essence is not the same as person. For whenever essence is the same as person or "suppositum," there can be only one "suppositum" of one nature, as is clear in the case of all separate substances. For in those things which are really one and the same, one cannot be multiplied apart from the other. But in God there is one essence and three persons, as is clear from what is above expounded (28, 3; 30, 2). Therefore essence is not the same as person.
Objection 2. Further, simultaneous affirmation and negation of the same things in the same respect cannot be true. But affirmation and negation are true of essence and of person. For person is distinct, whereas essence is not. Therefore person and essence are not the same.
Objection 3. Further, nothing can be subject to itself. But person is subject to essence; whence it is called "suppositum" or "hypostasis." Therefore person is not the same as essence.
On the contrary, Augustine says (De Trin. vi, 7): "When we say the person of the Father we mean nothing else but the substance of the Father."
I answer that, The truth of this question is quite clear if we consider the divine simplicity. For it was shown above (Question 3, Article 3) that the divine simplicity requires that in God essence is the same as "suppositum," which in intellectual substances is nothing else than person. But a difficulty seems to arise from the fact that while the divine persons are multiplied, the essence nevertheless retains its unity. And because, as Boethius says (De Trin. i), "relation multiplies the Trinity of persons," some have thought that in God essence and person differ, forasmuch as they held the relations to be "adjacent"; considering only in the relations the idea of "reference to another," and not the relations as realities. But as it was shown above (Question 28, Article 2) in creatures relations are accidental, whereas in God they are the divine essence itself. Thence it follows that in God essence is not really distinct from person; and yet that the persons are really distinguished from each other. For person, as above stated (29, 4), signifies relation as subsisting in the divine nature. But relation as referred to the essence does not differ therefrom really, but only in our way of thinking; while as referred to an opposite relation, it has a real distinction by virtue of that opposition. Thus there are one essence and three persons.
Reply to Objection 1. There cannot be a distinction of "suppositum" in creatures by means of relations, but only by essential principles; because in creatures relations are not subsistent. But in God relations are subsistent, and so by reason of the opposition between them they distinguish the "supposita"; and yet the essence is not distinguished, because the relations themselves are not distinguished from each other so far as they are identified with the essence.
Reply to Objection 2. As essence and person in God differ in our way of thinking, it follows that something can be denied of the one and affirmed of the other; and therefore, when we suppose the one, we need not suppose the other.
Reply to Objection 3. Divine things are named by us after the way of created things, as above explained (13, 1,3). And since created natures are individualized by matter which is the subject of the specific nature, it follows that individuals are called "subjects," "supposita," or "hypostases." So the divine persons are named "supposita" or "hypostases," but not as if there really existed any real "supposition" or "subjection."
Source: Summa Theologica (New Advent)