1. What is the difference between an Aristotelian primary substance and a supposit (hypostasis, suppositum)?
2. Is there any non-theological basis for this distinction?
3. If the answer to (2) is negative, is the addition of suppposita to one's Aristotelian ontology a case of legitimate metaphysical revision or a case of an ad hoc theoretical patch job? According to Marilyn McCord Adams, "Metaphysical revision differs from ad hoc theoretical patching insofar as it attempts to make the new data systematically unsurprising in a wider theoretical context." ("Substance and Supposits," p. 40)
'Supposit' is originally a logical term, whereas 'substance' is a metaphysical one. ('Hypostasis' is trickier, since it originally had to do double-duty.) The suppositum is that for which one's term stands. Originally only subject terms were taken to have supposition, although this was expanded later to allow for a kind of supposition in the predicate. Substances would have usually been taken to be paradigmatic supposita, although it's very clear that our subject terms can supposit for things other than substances. Thus the distinction between the two is easy enough to establish. 'Suppositum' is simply something of which something else may be predicated. Substances are the most obvious supposita, but we all have often taken concepts in the mind, abstract qualities, and bits of language as supposita. What counts as a suppositum will simply depend on the context.
While he doesn't always use the term, we see exactly this distinction between the logical and the metaphysical in Thomas Aquinas's Christology. A considerable part of what St. Thomas says on the subject is reducible to a few rules that give what might be called the grammar of orthodox Christology:
(1) We distinguish the natures by reason of which things are predicated of Christ.
(2) The natures, being distinct, are not predicated of each other in the abstract.
(3) We do not distinguish that which is predicated of Christ as suppositum.
(4) The natures, agreeing in suppositum, are predicated of each other in the concrete.
(5) A term in the predicate is taken formally, for the nature.
(6) A term in the subject is taken materially, for the suppositum.
(7) Contraries are only predicated of Christ in different natures.
(8) Things of which we may doubt to what nature they belong are only to be predicated with qualification.
All of these rules are explicitly appealed to by St. Thomas in order to resolve problems. (We can add rules for reduplication, but they are arguably reducible to (5) and (6).) They are simply logical rules for how to speak given a basic assumption, that Christ is one divine person in two natures, one human and one divine. Given this, we know that it is false that 'Christ became creature' if taken in an unqualified way (by 8), that 'Christ as man is creature' is true (by 5), that 'Christ as man is God' is false (by 6); and so forth. Thus note that what goes with suppositum is predicate. However, predicate terms are generally taken, where there is no special qualification, to indicate natures or properties of natures, and subject terms are in a similar way generally taken to supposit for the person.
As for what the supposita are, metaphysically, in the case of the Trinity and the Incarnation, Aquinas thinks we can know some things, but it's all indirect. That is, we don't have any direct understanding of what a divine person is, although from our indirect information (i.e., divine testimony) we can say some things about why (for instance) it's correct to use the word 'person' to name it.