How do the Father and Son differ if they have the same particular essence? My question assumes ousia in Christian orthodoxy means a particular (not universal) essence or form, and not a concrete instantiation (hypostasis).
If the Father and Son have the same ousia, where are the properties that distinguish them? Does the hypostasis have additional distinguishing properties? If so, are these properties necessary or contingent? If necessary, how are they definitionally excluded from the ousia?
I think the traditional Trinitarian answer would be that (prior to the incarnation) the three persons of the Trinity do not differ by any properties; instead they are distinguished by their relationships with each other. The three persons of the Trinity all share the one singular divine essence. Faculties such as the will and mind belong to the divine essence, so there is one single divine will which each person possesses, one single divine mind which each person possesses. Attributes like God's omniscience, omnibenevolence, justice, impassibility, transcendence, these are all attributes of the divine essence.
Without implying that the Son is in any way non-eternal, created, or subservient, the Father is first, and the Son comes from the Father; he is the eternally begotten Son. Likewise the Spirit comes from the Father (and possibly the Son, depending on who you ask). Likewise, in their relationships with each other, the persons of the Trinity cooperate in their creation of the world, and in their salvation of humanity, but in their cooperation they each assume a particular role. This has been called the Economic Trinity. But the traditional Trinitarian position is not that we should think the roles each person assumed are arbitrary or interchangeable; though we do not fully understand how the persons of God relate with each other, it was not chance that the Son became incarnate, but it is infinitely fitting that the Son, as the Word of the Father, would become the incarnate Word to fully reveal God to us. Likewise it is fitting that the Spirit of God would indwell us, breathing new spiritual life into us. (Spirit and breath are one word in both Hebrew and Greek.)