In the fourth and final article on the circumstances of human acts, St. Thomas explains why St. Gregory of Nyssa was right to say that "why" and "what" are the two most important circumstances. (Okay, technically he's explaining why Aristotle was right to say the same thing. But St. Gregory provides the sed contra.)
An act is human insofar as it is voluntary, and it is voluntary insofar as the will is freely moved, and the will is freely moved insofar as there is some end it desires. So the circumstance that touches on the end of the act is the one that is most closely linked to what makes the act a human act, and that circumstance is the "why."
Subsequent tradition has certainly agreed with the two Doctors. Philosophically speaking, the intention of an act is a circumstance, but in moral theology intention is promoted to a distinct component of the human act, separate from the rest of the circumstances. (For that matter, the third component, the object, is also an end, albeit one that, as St. Thomas mentioned in Article 3 ad 3, is a condition and not a circumstance.)
Second only to the end in importance is what most directly touches the substance of the act, which per St. Thomas is the "what" of the act. Again, this may be better thought of as "what is effected," so as not to confuse it with e.g. the "what we do" that is the objective component of the act in the U.S. Catholic Catechism for Adults. The importance of the circumstance "what" in moral theology is reflected no least in the importance of the principle of double effect, according to which a single unintended bad effect can put the kibosh on an act regardless of other, less important (i.e., non-end-related and non-effect-related) circumstances.
As for those other, less important circumstances, for a given act they are ranked in importance based on how closely they approach the end or the substance of the act.
The objections to this ranking seem a bit pro-forma:
When Aristotle say that "in which the act is" is the second-most important circumstance, doesn't he mean "time and place," which aren't very important at all? No, he means "what."
The end is altogether outside the act. How can it be the most important circumstance? Because it's the most important cause of the act.
Aren't "who" and "mode of acting" the cause and form of the act, in which case aren't they the most important circumstances? No, in which case no.
Okay, the final reply is a little more interesting than that: While the "who" is the primary efficient cause, the efficient cause is not as important as the final cause; moreover, "who" includes a lot of circumstances that aren't very important to the act at all (which is not true of "why").
Meanwhile, it seems that students commonly took "mode of acting" to mean something more than it does. As was explained back in Article 2, it only adds certain accidental qualities to an act, and as such is not of major importance.