To borrow a statement from Benedict Ashley’s essay in Philosophy and the God of Abraham: What does Aristotelian Thomism teach? Here is a brief summary in 8 sentences:
- “[T]he philosophy of Aquinas, as distinct from his theology, is best gathered not from the Summa Theologiae (supplemented by the Commentary on the Sentences and the Summa Contra Gentiles, etc.), as Gilson for example chose to do, but from the commentaries on Aristotle, in which the philosophical disciplines are treated according to their own principles and methods via inventionis.
- Aquinas ought to be interpreted as a convinced Aristotelian who vigorously opposes every tendency to Platonize in epistemology, and admits Platonic elements into this thought from the Church Fathers only in so far as he can validate them in accordance with Aristotelian epistemology.
- [A] correct interpretation of Aquinas’ philosophy depends on a careful observance of his theory of the order of the sciences.
- [T]he many attempts… to distinguish the natural sciences as empirical from philosophy as ‘rational’ cannot be admitted in authentic Thomism.
- [T]he key to reading Aristotle and Aquinas on natural science is a good understanding of the Organon and especially the Posterior Analytics (which by the way deals with the kind of questions which today are commonly called ‘philosophy of science’).
- [T]he strikingly apparent differences between natural science as it developed after Galileo and as Aquinas conceived it are not due to any formal difference in the kind of knowledge which modern science achieves, but are due to the confused self-understanding of modern science which resulted from its ideological history after Galileo.
- [T]he natural science of Aristotle and Aquinas, no matter how obsolete in its details, still can provide modern science with the foundational analysis which can resolve the many paradoxes in which it now is bound up in intellectual incoherence and which have led to disastrous cultural and ethical results.
- [T]his task of revising modern science on the basis of its original foundations cannot be evaded by a flight to metaphysics or theology.” 
Aquinas’ Utilization of Aristotelian Metaphysics
Upon Thomas Aquinas’ commentary on chapter 2 of Boethius’ De Trinitate (Expositio super Librum Boethii de Trinitate), we find the following text from Boethius:
“Come then, let us enter into each matter, discussing it so it can be grasped and understood, for it seems well said that educated people try for such certainty as the matter allows. For since theoretical science can be divided into three –
- Natural Science, changing, non-abstract, unseparated: which considers bodily forms as forms-in-matter, forms that cannot existed separated from bodies that are changing – like falling earth or rising fire – so that what changes is the form-in-matter.
- Mathematics, changeless, non-abstract: which conceived bodily forms apart from matter and thus from change, though those forms exist in matter and so cannot be separated from matter and change.
- Theology, changeless, abstract, and inseparable: for God’s being lacks both matter and change –
in natural science we make use of reason, in mathematics discipline, in divine science intellect, not relying on imagination but rather scrutinizing Form itself [true form and no image, Existence itself from which all existence exists].” 
Aquinas asks (Q. 5, A. I) if whether or not this three-way distinction in legitimate. Arguing in the affirmative, Aquinas supplements his all too significant reply of the relationship and distinctions between the theoretical sciences:
“Now to be an object of our ability to speculate a thing must suit both our basic ability to understand and that further disposition to know things with certainty which perfects our understanding with science. The ability to understand, being itself non-material, requires its objects to be non-material; and the disposition to science requires its objects to be necessary – since we can know with certainty only what must be so.
What must be so is, as such, unchanging, since what changes, as such, can be or not be – exist or not exist in the unqualified sense, or be or not be the way it is. Any objects of speculation or theoretical science then, must, as such, position themselves in relation to matter and change, and according to their degrees of distance from matter and change such objects will determine different theoretical sciences.
Some objects of our speculation, then, depend on matter for their existence, unable to exist accept in matter. But a distinction is possible. Some depend on matter not only for existence but for understanding: since perceptible matter is included in their definition they cant be understood without that perceptible matter. Thus flesh and bone are included within the definition of a human being. Such things are studied by physics or natural science. Other things depend on matter for their existence but not for understanding, since perceptible matter is not included in their definition: mathematics.
Then there are other objects of speculation which don’t depend for their existence on matter and are able to exist out of matter, either never existing in matter, like God and the angels, or sometimes existing in matter but sometimes not, like substance, qualities, potentiality and actuality, the one and the many, etc. All such things are studied by theology – science of the divine – so called metaphysics – ‘after physics’ – because we learn it after physics, only able to reach what can’t be sensed through what can. And again it is called first philosophy, since all other sciences come after it in the sense of deriving their first principles from it. 
The Five Types of Thomism
Edward Feser in his wonderful two part series “The Thomistic Tradition,”  though specifically part 1, makes some important distinctions among the Thomist circle of debate/discussion taking place today:
- Neo-Scholastic Thomism: The dominant tendency within Thomism in the first decades after the revival sparked by Leo’s encyclical, this approach is reflected in many of the manuals and textbooks widely in use in Roman Catholic colleges and seminaries before Vatican II. Due to its emphasis on following the interpretative tradition of the great commentators on Aquinas (such as Capreolus, Cajetan, and John of St. Thomas) and associated suspicion of attempts to synthesize Thomism with non-Thomistic categories and assumptions, it has also sometimes been labeled “Strict Observance Thomism.”
- Existential Thomism: Etienne Gilson (1884-1978), the key proponent of this approach to Thomism, tended to emphasize the importance of historical exegesis but also to deemphasize Aquinas’s continuity with the Aristotelian tradition, highlighting instead the originality of Aquinas’s doctrine of being or existence.
- Laval or River Forest Thomism: This approach emphasizes the Aristotelian foundations of Aquinas’s philosophy, and in particular the idea that the construction of a sound metaphysics must be preceded by a sound understanding of natural science, as interpreted in light of an Aristotelian philosophy of nature. Accordingly, it is keen to show that modern physical science can and should be given such an interpretation. approaches are not necessarily incompatible.)
- Transcendental Thomism: Unlike the first three schools mentioned, this approach, associated with Joseph Marechal (1878-1944), Karl Rahner (1904-84), and Bernard Lonergan (1904-84), does not oppose modern philosophy wholesale, but seeks to reconcile Thomism with a Cartesian subjectivist approach to knowledge in general, and Kantian epistemology in particular.
- Lublin Thomism: This approach, which derives its name from the University of Lublin in Poland where it was centered, is also sometimes called “phenomenological Thomism.” Like transcendental Thomism, it seeks to combine Thomism with certain elements of modern philosophy, though in a way that is less radically revisionist.
While I admire the historical scholarship of Etienne Gilson, I tend to side with the Laval camp of Thomism as it emphasizes the proper understanding and usage of Aristotelian metaphysics in order to understand the natural sciences. As a philosophy and theoretical physics double major, I appreciate and can resonate with the kind of distinction some River Forest Thomists place on Aquinas’ theology/metaphysics.
-  From Benedict Ashley, OP, “The River Forest School and the Philosophy of Nature Today,” in R. James Long’s Philosophy and the God of Abraham: Essays in Memory of James A. Weisheipl, OP (Pontifical Institute of Medieval Studies: 1991), pp. 1-13.
-  Quoted from Aquinas: Selected Philosophical Writings, trans. and ed. by Timothy McDermott (Oxford University Press: 1993), pp. 1-2.
-  Ibid., 8.
-  From Ed Feser, “The Thomist Tradition,” http://edwardfeser.blogspot.com/2009/10/thomistic-tradition-part-i.html?m=1#6717933511240612484
*The featured image to this article was taken from https://dogmatics.wordpress.com/2013/10/10/transubstantiation-in-thomas-aquinas-part-three/.
The original painting is Benozzo Gozzoli’s Triumph of St. Thomas Aquinas (1484).