A Brief Explanation of the Trinity - How are the three persons distinct?
They are distinct in their relationship with each other and in their function or role in creation and in the plan of redemption and salvation.
In the work of creation, the three persons of the Father, the Son, and the Holy Spirit had different functions. God the Father spoke the world into existence (Genesis 1-3), but God the Son carried out the divine decree. The Gospel of John says of Jesus: "All things were made through him [Jesus], and without him was not anything made that was made" (John 1:3). God the Holy Spirit was active in creation, "brooding over the face of the waters" (Genesis 1:2; see also Psalm 33:6 and 139:7).
In the work of redemption and salvation, the three persons have different roles or functions. God the Father planned the redemption and sent God the Son into the world to carry out the plan of redemption. He obeyed the Father and died on the cross for our sins. Neither the Father nor the God the Holy Spirit carried out this part of redemption. The Holy Spirit was sent by the Father and the Son (John 14:26 and 16:7) to apply the plan of redemption to the heart of each person who receives it. The Holy Spirit also purifies and sanctifies us or makes us holy in our daily lives.
In relation to each other, we see that in the creation and redemption, the Father sends and directs the Son (Ephesians 3:14-15), who obeys and goes where he is directed, revealing the full nature of God the Father to us (John 1:1-5, 14, 18; 17:4; Philippians 2:5-11). These roles fit perfectly with Fatherhood and Sonship. Both the Father and the Son send the Holy Spirit, so the Spirit obeys both the first and second persons of the Trinity.
Thus, while the three persons of the Trinity have and share the same attributes (holiness, mercy, omniscience, and so on), they have distinct roles or functions as they relate to each other, to creation, and in the plan of redemption and salvation. They are equal to each other in their divine attributes, but the Son and the Spirit are subordinate in their roles.
For more on these ideas, see Wayne Grudem, Systematic Theology, Inter-varsity Press (UK) and Zondervan (US), 2000.