For more information, see: Lagrange’s The One God: A Commentary on Prima Pars, Reality: A Synthesis of Thomistic Thought; Sacrae Theologiae Summa IIA, Appendix; Nicoloas’ Catholic Dogmatic Theology: A Synthesis, Volume 1.
The online Orthodox are often quick to cast judgment on the Doctrine of God of their Latin brethren as somehow deficient. Yet, the opposite is the case. Rather than expressing a complete break in the person of St. Augustine, Latin theology provides a synthesis and organic development on the Trinitarian theology of the Greek Fathers, which allows for a greater depth and clarity in Latin authors.
First, a brief misconception must be dispelled. It was popular in early 20th century historical theology to speak of a very strict difference between the Latins and Greeks, along the line of such, “Latin philosophy first of all envisions the nature in itself and proceeds to the supposit. Greek philosophy first of all envisions the supposit and then penetrates it so as to find the nature. The Latin thinker considers personality as a mode of the nature.” 
From this it is concluded, “Therefore the Latins were inclined to search for the reason of the processions in the divine nature and in its immanent activities; here we find the origin of the psychological theory about the divine processions, started by St. Augustine1 2 and brought to perfection through S. Anselm by St. Thomas. Among the Greeks however none of this is found and the processions are not thought of as activities of the nature but as donations…in reference to ad extra activities the Latins hold such a strict unity of the whole acting Trinity that they reduce the diversity of attributions, which occur in the sources, to a mere appropriation; in contrast, according to the Greeks the individual persons act in a way proper to themselves, which cannot be reduced to an appropriation.” 
The Reason behind this myth was (usually) to discredit Latin theology and prefer Greek theology. Namely, these authors (generally) sought to eliminate from Trinitarian theology, 1. Psychological analogies of the Trinity, and 2. Hypostatic appropriations, i.e., that when Sacred Scripture speaks of a certain person of the Trinity doing x, it is truly the one and undivided Trinity acting, and such reference is a mere appropriation.
To the principle, we may answer, “From the perspective of Trinitarian theology, the schema drawn from the artificial opposition between the respective philosophies of the person and that of the nature is extremely simplistic. The Greeks would first of all consider the Three Persons in their distinction, and the Latins would base their reflections on the one nature. This is to forget that the Greeks founded their reflection on the Bible, which proclaims God’s unity before speaking (in the New Testament) about the distinct Persons. In reality, for every theologian, the Three Persons together are the One God. He who considers the distinction of the Persons must at the same time consider the unity of the Nature, for the one, indivisible Divine Nature is what the Father communicates to the Son, and the Son and the Father to the Holy Spirit. He who considers the Nature must at the same time consider the distinction of Persons, for the one Nature subsists in three distinct Persons. No theology could privilege one of the two considerations to the detriment of the other without simultaneously distorting the consideration that he intends to privilege.” 
To the two conclusions given, it is simply not the case that the Greek fathers are ignorant of such. Regarding appropriations, St. Basil writes, “He sanctifies and vivifies and enlightens and consoles and all things like that are done equally by the Father and the Son and the Holy Spirit....Therefore the identity of action in the Father and the Son and the Holy Spirit clearly shows the absolute likeness of nature.”
Regarding Psychological analogies, St. John of Damascus writes, “since God is everlasting and perfect, He will have His Word subsistent in Him, and everlasting and living, and possessed of all the attributes of the Begetter. For just as our word, proceeding as it does out of the mind, is neither wholly identical with the mind nor utterly diverse from it (for so far as it proceeds out of the mind it is different from it, while so far as it reveals the mind, it is no longer absolutely diverse from the mind, but being one in nature with the mind, it is yet to the subject diverse from it), so in the same manner also the Word of God in its independent subsistence is differentiated from Him from Whom it derives its subsistence: but inasmuch as it displays in itself the same attributes as are seen in God, it is of the same nature as God. For just as absolute perfection is contemplated in the Father, so also is it contemplated in the Word that is begotten of Him.” 
Now that we have gotten that out of the way, let’s get to the true superiority of Latin theology in the realm of Trinitarian theology. Latin Trinitarian theology truly shines because of the Augustinian synthesis given in De Trinitate. This chiefly is found because of theological method, while the Greek Fathers (especially St. John of Damascus) reached many profound speculations, the Greeks in their classical texts were bound more by polemics against heretics than they were with creating systems.
This is exemplified in the fact that Latins were able to establish with greater depth and clarity the “formal principle” of the processions as subsisting relations of volition and intellection (while the Greeks were able to reach these ideas moreso in their general outlines). Generally, this allowed the Latins (beginning with St. Augustine in De Trinitate, through St. Anselm and Lombard, and to St. Thomas) to answer 10 questions that the Greeks did not generally answer (in some cases, these answers were exclusively Latin).
First, what is the manner of the begetting of the Son? Fr. Lagrange writes, “St. Augustine explains the intimate manner of the generation of the eternal and only-begotten Son, while the Greek Fathers said that the manner of His begetting was inscrutable. Explaining the prologue of St. John's Gospel, St. Augustine showed that the Father from eternity begets His Son by an intellectual act just as our mind conceives the mental word: in the soul we find the mind, knowledge, and love; in the soul, which is the image of the Trinity, there are memory, intelligence (the act of intellection), and the will.” 
Second, what is the manner of the spiration of the Spirit? The same author writes, “The manner of the second procession, which appears as the procession of love, is also explained. From our souls, which according to the Scriptures are created in the likeness of God, proceeds not only the word but also love. The human mind not only conceives the true-good but also loves it. If therefore the only-begotten Son proceeds from the Father as the mental Word, the Holy Ghost is to be considered as proceeding from them as love.”
Third, why are there only two processions? This is explained when the formal constituents of the processions are found in immanent psychological acts. For, only intellection and volition can be said to be immanent, therefore the conclusion follows. As St. Thomas writes, “The divine processions can be derived only from the actions which remain within the agent. In a nature which is intellectual, and in the divine nature, these actions are two, the acts of intelligence and of will. The act of sensation, which also appears to be an operation within the agent, takes place outside the intellectual nature, nor can it be reckoned as wholly removed from the sphere of external actions.” 
Fourth, why is the first procession called generation? This is something which St. Augustine himself could not explain, yet the later Latin theologians explained using his principles. St. Thomas writes, “So in this manner the procession of the Word in God is generation; for He proceeds by way of intelligible action, which is a vital operation:—from a conjoined principle (as above described):—by way of similitude, inasmuch as the concept of the intellect is a likeness of the object conceived:—and exists in the same nature, because in God the act of understanding and His existence are the same, as shown above.” 
Fifth, how do the two processions differ? This is explained by the concept of mutual oppositions of subsisting relations (along with the different manner of intellection and of volition). For, the Son differs from the Spirit in that the Son stands in an opposite relation to the Father begetting/begotten, and the Spirit stands in an opposite relation to BOTH the Father and the Son spirating/spirated.
Sixth, why is the second procession not called generation? St. Thomas writes, “The procession of love in God ought not to be called generation. In evidence whereof we must consider that the intellect and the will differ in this respect, that the intellect is made actual by the object understood residing according to its own likeness in the intellect; whereas the will is made actual, not by any similitude of the object willed within it, but by its having a certain inclination to the thing willed. Thus the procession of the intellect is by way of similitude, and is called generation, because every generator begets its own like; whereas the procession of the will is not by way of similitude, but rather by way of impulse and movement towards an object.” 
Seventh, why does the Spirit proceed from the Father and the Son? Fr. Lagrange writes, “in our souls love proceeds not only from the soul itself but from the knowledge of the true-good, since nothing is loved unless it is also known.”
Eighth, since each of the persons share the one Divine intellect, why does not each person produce another Son, ad infinitum? Fr. Lagrange writes, “This difficulty is solved by the distinction between intellection and the expression of the notional idea inasmuch as the three persons all have intelligence but only the Father expresses the intellection. He alone expresses because the Word is adequate and the most perfect expression of the divine nature and no other Word need be enunciated. Just as in a classroom while the teacher is teaching, both he and the pupils understand, but the teacher alone enunciates.”
Ninth, a similar question, since each of the persons share the one Divine will, why does not each person produce another Spirit, ad infinitum? Fr. Lagrange writes, “The solution of this difficulty depends on the distinction between essential love, which is common to the three persons, and notional love, which is active spiration and corresponds to the enunciation of the Word. It is called notional because it denotes the third person. Thus the three persons all love, but only the first two spirate. We have then three kinds of love in God: essential, notional, and personal. Personal love is the Holy Ghost Himself, who is the terminus of active spiration just as the Word is the terminus of generation and enunciation. According to a rather remote analogy: a saintly preacher loves God and inspires his audience with this love, and the hearers also love God but they do not inspire others with this love.”
Tenth, why can't the Trinity be known from reason alone? The answer given by the Latins is that the Trinity cannot be known from creation because creation is an act of the One Godhead according to nature which only reveals the Divine unity, excepting the "footprints" found in the nature of Being, and the image found in the Soul (see: The Trinity Revealed in Creation where I write about this in length).
 Théodore de Régnon, Études de théologie positive sur la Sainte Trinité, 4 vols. (Paris: Victor Retaux, 1892–1898), vol. 1, 433.
 STS IIA, Appendix.
 Nicoloas, 123.
 John Damascene, “An Exact Exposition of the Orthodox Faith,” in St. Hilary of Poitiers, John of Damascus, ed. Philip Schaff and Henry Wace, trans. S. D. F. Salmond, vol. 9b of A Select Library of the Nicene and Post-Nicene Fathers of the Christian Church, Second Series (New York: Christian Literature Company, 1899), 5.
 n.b., such is not the case with St. John of Damascus as quoted above.