All people possess a human nature, which includes attributes such as a mind, a body, and a will. So, every human being is fully human regardless of gender, race, age, or other qualities. But even though every human being possesses a human nature, we have distinct attributes. For instance, the mind of one human being is not the same as the mind of another. The same applies for every essential human attribute. We are all fully human, but there are differences between us in terms of our human attributes. We do not share the same mind, soul, body, will, or any other human attribute.
The unity of the divine persons with respect to the divine nature is wholly different. The three persons of the Trinity are all divine in that to have a divine nature is to be divine; yet, the divine attributes are also identical between the three persons of the Godhead. The Father, Son, and Holy Spirit do not have three different minds; They have one identical mind. The same applies for every other divine attribute.
What makes the three persons of the Trinity differ from one another is a difference in relations, not in attributes. From the early church fathers through the Protestant Reformers to today, orthodox Christianity has said that what makes the Father the Father is that He is eternally unbegotten and what makes the Son the Son is that He is eternally begotten. Evidence for this is found in passages such as John 1:18, which in the KJV refers to the Son as "begotten." This is a better translation of the Greek than in some newer English versions, for the underlying Greek word has to do with generation. The Son is eternally generated by the Father. This generation, or begottenness, never had a beginning. The Son has always existed and has always been fully God even though He is begotten of the Father. And the Father has always begotten the Son such that the Son and the Father are both fully God.
Unbegottenness is the unique personal property of the Father, begottenness is the unique personal property of the Son, and procession is the unique personal property of the Holy Spirit. Following such texts as John 14:26, we say that the Holy Spirit proceeds from the Father and the Son. This passage, Dr. R.C. Sproul explains in his commentary John, says that "the Spirit is sent by both Father and Son," and Christian thinkers have seen this sending as reflective of a relationship of the Holy Spirit to Father and Son before time began. The Father and the Son sent the Holy Spirit at Pentecost because He has proceeded from Them from all eternity.
Why should we care to know these things about divine unity and distinction? It is not merely to fill our heads with theological knowledge. These concepts give us a further glimpse at who God is, allowing us to be filled with awe at how much greater God is than we can imagine. Knowing His set-apartness moves us to worship Him for His greatness, and thus we fulfill the purpose for which we were made.