Objection 1. It would seem that the Son is not other than the Father. For "other" is a relative term implying diversity of substance. If, then, the Son is other than the Father, He must be different from the Father; which is contrary to what Augustine says (De Trin. vii), that when we speak of three persons, "we do not mean to imply diversity."
Objection 2. Further, whosoever are other from one another, differ in some way from one another. Therefore, if the Son is other than the Father, it follows that He differs from the Father; which is against what Ambrose says (De Fide i), that "the Father and the Son are one in Godhead; nor is there any difference in substance between them, nor any diversity."
Objection 3. Further, the term alien is taken from "alius" [other]. But the Son is not alien from the Father, for Hilary says (De Trin. vii) that "in the divine persons there is nothing diverse, nothing alien, nothing separable." Therefore the Son is not other that the Father.
Objection 4. Further, the terms "other person" and "other thing" [alius et aliud] have the same meaning, differing only in gender. So if the Son is another person from the Father, it follows that the Son is a thing apart from the Father.
On the contrary, Augustine [Fulgentius, De Fide ad Petrum i.] says: "There is one essence of the Father and Son and Holy Ghost, in which the Father is not one thing, the Son another, and the Holy Ghost another; although the Father is one person, the Son another, and the Holy Ghost another."
I answer that, Since as Jerome remarks [In substance, Ep. lvii.], a heresy arises from words wrongly used, when we speak of the Trinity we must proceed with care and with befitting modesty; because, as Augustine says (De Trin. i, 3), "nowhere is error more harmful, the quest more toilsome, the finding more fruitful." Now, in treating of the Trinity, we must beware of two opposite errors, and proceed cautiously between them--namely, the error of Arius, who placed a Trinity of substance with the Trinity of persons; and the error of Sabellius, who placed unity of person with the unity of essence.
Thus, to avoid the error of Arius we must shun the use of the terms diversity and difference in God, lest we take away the unity of essence: we may, however, use the term "distinction" on account of the relative opposition. Hence whenever we find terms of "diversity" or "difference" of Persons used in an authentic work, these terms of "diversity" or "difference" are taken to mean "distinction." But lest the simplicity and singleness of the divine essence be taken away, the terms "separation" and "division," which belong to the parts of a whole, are to be avoided: and lest quality be taken away, we avoid the use of the term "disparity": and lest we remove similitude, we avoid the terms "alien" and "discrepant." For Ambrose says (De Fide i) that "in the Father and the Son there is no discrepancy, but one Godhead": and according to Hilary, as quoted above, "in God there is nothing alien, nothing separable."
To avoid the heresy of Sabellius, we must shun the term "singularity," lest we take away the communicability of the divine essence. Hence Hilary says (De Trin. vii): "It is sacrilege to assert that the Father and the Son are separate in Godhead." We must avoid the adjective "only" [unici] lest we take away the number of persons. Hence Hilary says in the same book: "We exclude from God the idea of singularity or uniqueness." Nevertheless, we say "the only Son," for in God there is no plurality of Sons. Yet, we do not say "the only God," for the Deity is common to several. We avoid the word "confused," lest we take away from the Persons the order of their nature. Hence Ambrose says (De Fide i): "What is one is not confused; and there is no multiplicity where there is no difference." The word "solitary" is also to be avoided, lest we take away the society of the three persons; for, as Hilary says (De Trin. iv), "We confess neither a solitary nor a diverse God."
This word "other" [alius], however, in the masculine sense, means only a distinction of "suppositum"; and hence we can properly say that "the Son is other than the Father," because He is another "suppositum" of the divine nature, as He is another person and another hypostasis.
Reply to Objection 1. "Other," being like the name of a particular thing, refers to the "suppositum"; and so, there is sufficient reason for using it, where there is a distinct substance in the sense of hypostasis or person. But diversity requires a distinct substance in the sense of essence. Thus we cannot say that the Son is diverse from the Father, although He is another.
Reply to Objection 2. "Difference" implies distinction of form. There is one form in God, as appears from the text, "Who, when He was in the form of God" (Philippians 2:6). Therefore the term "difference" does not properly apply to God, as appears from the authority quoted. Yet, Damascene (De Fide Orth. i, 5) employs the term "difference" in the divine persons, as meaning that the relative property is signified by way of form. Hence he says that the hypostases do not differ from each other in substance, but according to determinate properties. But "difference" is taken for "distinction," as above stated.
Reply to Objection 3. The term "alien" means what is extraneous and dissimilar; which is not expressed by the term "other" [alius]; and therefore we say that the Son is "other" than the Father, but not that He is anything "alien."
Reply to Objection 4. The neuter gender is formless; whereas the masculine is formed and distinct; and so is the feminine. So the common essence is properly and aptly expressed by the neuter gender, but by the masculine and feminine is expressed the determined subject in the common nature. Hence also in human affairs, if we ask, Who is this man? we answer, Socrates, which is the name of the "suppositum"; whereas, if we ask, What is he? we reply, A rational and mortal animal. So, because in God distinction is by the persons, and not by the essence, we say that the Father is other than the Son, but not something else; while conversely we say that they are one thing, but not one person.
Source: Summa Theologica (New Advent)