How distinct are the Persons of the Trinity - the Father, Son and Holy Spirit?

Are there three distinct Persons in the Godhead, to be distinguished from one another?
It is wise to be humble as we approach the high and holy subject of the Being of God.  Any mere discussion of Him, before whom the angels veil their faces, the Three in One and the One in Three, should be guided by Him. We must not try to define or explain or measure where the Word of God is silent; in fact our wisdom is to limit ourselves as far as possible to the words of Scripture. We "cannot by searching find out God" (Job 11:7), except in so far as He is pleased to reveal Himself.

God has never condescended to prove His existence, nor yet explain His mode of Being; but He reveals Himself in His Works and Word; and even then we need God the Spirit to teach us. The great truth of the Old Testament is the Unity of the Godhead. “The Lord our God is one Lord” (Deuteronomy 6:4), and we believe this as strongly as any Jew or so-called Unitarian. But from the very first verse of Genesis we are made aware of a Plurality in that Unity. The word for “created” is in the singular, but “God” is a plural word— Elohim, though when this stands for strong ones or false gods, it takes a plural verb (e.g., see Exodus 32:4). In verse 2 a Personal Being in the Godhead is differentiated: “the Spirit of God.” Then in verse 26 we read, “Let us make man in our image.” Later, in Exodus we become aware of another Person—the Angel of the Lord, to whom Divine honours are paid (chap. 23.20-22)—and yet He is distinct from Jehovah. In Matthew 3:1 the word “messenger” (“messenger of the covenant”) is in the Hebrew the same as “angel”. We are brought face to face with a Personal Wisdom possessed by Jehovah “before His works of old” (Proverbs 8: 22-31); and in Proverbs 30:4 we are asked, Who is the Creator and “what is His Son’s name?” Who then is this Angel of His presence— and Who this Spirit against whom Israel rebelled and Who was grieved with their ways? (see also Zechariah 12:10 and Zechariah 13:7). Clearly even in the Old Testament there are in the Unity of the Godhead, three Personal Subsistencies.

In the New Testament this becomes clearer. In Matthew Jesus is identified with Immanuel—”God with us.” But “God so loved the world that He gave His only-begotten Son” is equally true, so that the Giver and the Given One are both God. Again in John 1:1 we read “The word was with (or in relation to—pros) God, and the Word was God” and in V. 14, “the Word became flesh.” To Him at His baptism the heavens were opened and He saw the Spirit of God descending like a dove and heard a voice from heaven saying, “This is my beloved Son in whom I sin well pleased” (Matthew 3:16-17). Later, He “by the Eternal Spirit offered Himself without spot to God” (Hebrews 9:14) and then after the Lord ascended, the Father sent the Spirit, in His and His own name, to testify of Christ.

All who believe are to be baptized into the Name of the Father and of the Son and of the Holy Ghost (Matthew 28.), and to such comes the message, “The grace of the Lord Jesus Christ and the love of God and the communion of the Holy Ghost be with you all” (2 Corinthians 13:14). “For through Him we both have access by one Spirit to the Father” (Ephesians 2:18). We do not confound the Persons nor do we divide the Substance. There is a distinction in the Divine Functions. The Father preordains—determines “the times and seasons,” and gives the Son; The Son creates, redeems, intercedes; The Spirit calls, gives life, indwells. But those three Divine Persons are coequal, coeternal, co-substantial in the Unity of the Godhead.