How do the Father and Son differ if they have the same particular essence?

The fathers believed that both species (ousia) and persons (hypostases) were substances (nature) in the Aristotelian sense. The former refers to the nature that is common to the persons (i.e. human nature) whilst the latter is the nature that is instantiated in a particular individual (e.g. human person). The what is common (the species) to particulars is the secondary substance and the particulars (the persons) is primary substance. Both species and genus are under secondary substance but the fathers regarded ousia as species, not genus, because genus refers to many species (= many ousias). For example, 'humans', 'dogs' and 'horses' are all different species ( = different ousias) under the same genus "animals". (source: On "Not Three Gods" - Again: Can a Primary-Secondary Substance Reading of Ousia and Hypostasis Avoid Tritheism? Nathan A. Jacobs, 2008)

When we count those who possess a singular human nature, we call them many human persons since persons are what is plural, not the nature. And we call them individually by their names for names signify the plurality of distinct persons. Peter, James and John were not three humans (i.e. three human natures) but three human persons (i.e. three persons with the same human nature). Likewise, the Father, Son and Holy Spirit are not three Gods but one God, due to their divine nature being one and the same. They're three divine persons (i.e. three persons with the same divine nature).

We say, then, to begin with, that the practice of calling those who are not divided in nature by the very name of their common nature in the plural, and saying they are "many men," is a customary abuse of language, and that it would be much the same thing to say they are "many human natures." And the truth of this we may see from the following instance. When we address any one, we do not call him by the name of his nature, in order that no confusion may result from the community of the name, as would happen if every one of those who hear it were to think that he himself was the person addressed, because the call is made not by the proper appellation but by the common name of their nature: but we separate him from the multitude by using that name which belongs to him as his own;-- that, I mean, which signifies the particular subject. Thus there are many who have shared in the nature--many disciples, say, or apostles, or martyrs--but the man in them all is one; since, as has been said, the term "man" does not belong to the nature of the individual as such, but to that which is common. For Luke is a man, or Stephen is a man; but it does not follow that if any one is a man he is therefore Luke or Stephen: but the idea of the persons admits of that separation which is made by the peculiar attributes considered in each severally, and when they are combined is presented to us by means of number; yet their nature is one, at union in itself, and an absolutely indivisible unit, not capable of increase by addition or of diminution by subtraction, but in its essence being and continually remaining one, inseparable even though it appear in plurality, continuous, complete, and not divided with the individuals who participate in it. And as we speak of a people, or a mob, or an army, or an assembly in the singular in every case, while each of these is conceived as being in plurality, so according to the more accurate expression, "man" would be said to be one, even though those who are exhibited to us in the same nature make up a plurality. Thus it would be much better to correct our erroneous habit, so as no longer to extend to a plurality the name of the nature, than by our bondage to habit to transfer to our statements concerning God the error which exists in the above case. But since the correction of the habit is impracticable (for how could you persuade any one not to speak of those who are exhibited in the same nature as "many men"?--indeed, in every case habit is a thing hard to change), we are not so far wrong in not going contrary to the prevailing habit in the case of the lower nature, since no harm results from the mistaken use of the name: but in the case of the statement concerning the Divine nature the various use of terms is no longer so free from danger: for that which is of small account is in these subjects no longer a small matter. Therefore we must confess one God, according to the testimony of Scripture, "Hear, O Israel, the Lord thy God is one Lord," even though the name of Godhead extends through the Holy Trinity. This I say according to the account we have given in the case of human nature, in which we have learnt that it is improper to extend the name of the nature by the mark of plurality. We must, however, more carefully examine the name of "Godhead," in order to obtain, by means of the significance involved in the word, some help towards clearing up the question before us. (On Not Three Gods, to Ablabius Gregory of Nyssa)

Summary

The 4th century fathers spoke of "God" and "man" as referring to "nature" which is 'singular'. Their point is that we cannot say "many Gods" or "many men" because divine nature and human nature are both singular and the singular nature does not divide or become many when individuated.

The Father and the Son and the Holy Spirit have the "same ousia" (= secondary ousia, which is in this case, species, not genus) whilst each is a "particular ousia" ( = primary ousia or hypostasis/person). Both species and persons are equally substances ('nature' or 'properties') but they differ in the sense that the former refers to what is common (the species or properties/characteristics shared by many) whilst the latter refers to what is particular (an individual in which the properties/characteristics of species subsist).