The Father and the Son are different persons

Jesus states in many scriptures that He and the Father are quite different:

John 17:5 (King James Version)

5 And now, O Father, glorify thou me with thine own self with the glory which I had with thee before the world was.

Below (the LORD) denotes: The name of God, YHWH (Hebrew) known as Jehovah (English)

Jesus himself referred to his Father as “the only true God.”

(John 17:3) (The LORD) himself said: “Besides me there is no God.”

(Isa. 44:6) The apostle Paul wrote that, to true Christians, “there is . . . one God the Father.”

(1 Cor. 8:5, 6) So God is unique; no one else shares his position. God stands in utter contrast to all such objects of worship as idols, deified humans, and Satan. All these are false gods.

Jesus is spoken of in the Scriptures as “a god,” even as “Mighty God.” (John 1:1; Isa. 9:6)

But nowhere is he spoken of as being Almighty, as (The LORD) is. (Gen. 17:1)

Jesus is said to be “the reflection of [God’s] glory,” but the Father is the Source of that glory. (Heb. 1:3)

Jesus in no way seeks the position of his Father. He said: “It is (The LORD) your God you must worship, and it is to him alone you must render sacred service.” (Luke 4:8)

He exists “in God’s form,” and the Father has commanded that “in the name of Jesus every knee should bend,” but this is all done “to the glory of God the Father.”— Phil. 2:5-11

God spoke to his son after his baptism

Matthew 3:16-17 (American Standard Version)

16 And Jesus when he was baptized, went up straightway from the water: and lo, the heavens were opened unto him, and he saw the Spirit of God descending as a dove, and coming upon him;17 and lo, a voice out of the heavens, saying, This is my beloved Son, in whom I am well pleased.

One scripture that shows Jesus' origin, might shed some light:

Colossians 1:13-16 (King James Version)

13 Who hath delivered us from the power of darkness, and hath translated us into the kingdom of his dear Son: 14 In whom we have redemption through his blood, even the forgiveness of sins: 15 Who is the image of the invisible God, the firstborn of every creature: 16 For by him were all things created, that are in heaven, and that are in earth, visible and invisible, whether they be thrones, or dominions, or principalities, or powers: all things were created by him, and for him:

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The Son and the Holy Spirit are different persons; and the Holy Spirit and the Father are different persons?

The Hebrew word ruʹach and the Greek word pneuʹma, often translated “spirit,” have a number of meanings. All of them refer to that which is invisible to human sight and gives evidence of force in motion. The Hebrew and Greek words are used with reference to (1) wind, (2) the active life-force in earthly creatures, (3) the impelling force that issues from a person’s figurative heart and causes him to say and do things in a certain way, (4) inspired expressions originating from an invisible source, (5) spirit persons, and (6) God’s active force, or holy spirit

The Greek pneuʹma (spirit) comes from pneʹo, meaning “breathe or blow,” and the Hebrew ruʹach (spirit) is believed to come from a root having the same meaning. Ruʹach and pneuʹma, then, basically mean “breath” but have extended meanings beyond that basic sense. (Compare Hab 2:19; Re 13:15.) They can also mean wind; the vital force in living creatures; one’s spirit; spirit persons, including God and his angelic creatures; and God’s active force, or holy spirit.

Compare References Koehler and Baumgartner’s Lexicon in Veteris Testamenti Libros, Leiden, 1958, pp. 877-879; -Brown, Driver, and Briggs’ Hebrew and English Lexicon of the Old Testament, 1980, pp. 924-926; -Theological Dictionary of the New Testament, edited by G. Friedrich, translated by G. Bromiley, 1971, Vol. VI, pp. 332-451.

All these meanings have something in common: They all refer to that which is invisible to human sight and which gives evidence of force in motion. Such invisible force is capable of producing visible effects.

Probably the shortest answer to distinct persons, once the Trinity is already established as per your post, and that also includes a philosophical dimension rather than only scriptural authority, is

"39 And he went a little further, and fell on his face, and prayed, saying, O my Father, if it be possible, let this cup pass from me: nevertheless not as I will, but as thou wilt."

..and by extension to the Holy Spirit.

'not my will' is the strongest indication of concretely different persons in the case where someone might argue that the 'aspects' of the trinity are not a substantial reality but rather different views of the one god or some such other argument, arguing that Jesus is merely simplifying things for his simple human hearers.

Nevertheless, this doesn't prove it though proof isn't what you asked for, as there is the whole issue of the fact that Jesus is said to have two wills but one person, the divine and the human wills. But in any case, the proofs of God (and the foundation of real faith) have always been God himself, not reasoning or scripture, so in actual practice that's not a problem. Ie, "God, are you really three distinct persons?" (the corollary to the question that so often results in conversion "God, if you exist please reveal yourself"). [This presumes the reader knows that Christian faith is NOT defined as "Belief without evidence" which was a new definition presumptuously invented by an atheist, Bertrand Russell; rather almost the total opposite, with God providing himself as proof through union, if the person assents (with 'begging the question' resolved by the fact that we have a priori knowledge of God, "Made in the image", union then providing the 'likeness'.]

It's also worth mentioning, as it's a frequent issue, I can see even here, that the persons of the Trinity are distinct and not separate. Ie, a unity rather than a union. Which leads to the fun paradox: human persons are separate but not as distinct from each other as the persons of the Trinity, who are not separate. :)

A huge number of scriptures could be brought to bear as evidence for distinctions between the three persons of the godhead. I'll limit myself to following Louis Berkhof's treatment, which calls attention to several major ways that the persons are distinguished in Scripture.

To him, at least, through these passages "the separate persons of the Trinity are made to stand out clearly before our minds."

The Father sends the Son

Besides the ubiquitous John 3:16, Berkhof references Galatians 4:4, Hebrews 1:6, and 1 John 4:9. The beginning of Hebrews 1 in particular emphasizes that the Father sends the Son:

Long ago, at many times and in many ways, God spoke to our fathers by the prophets, ² but in these last days he has spoken to us by his Son, whom he appointed the heir of all things, through whom also he created the world. ³ He is the radiance of the glory of God and the exact imprint of his nature, and he upholds the universe by the word of his power. After making purification for sins, he sat down at the right hand of the Majesty on high, ⁴ having become as much superior to angels as the name he has inherited is more excellent than theirs.

⁵ For to which of the angels did God ever say,

“You are my Son,
    today I have begotten you”?

Or again,

“I will be to him a father,
    and he shall be to me a son”?

⁶ And again, when he brings the firstborn into the world, he says,

“Let all God's angels worship him.”

The Sending of the Spirit

Berkhof next mentions passages referring to the sending of the Holy Spirit, which he says is done by both Father and Son. Here he mentions John 14:26, John 15:26, John 16:7, and Galatians 4:6. Quoted below are John 14:26 and 16:7:

^14:26 But the Helper, the Holy Spirit, whom the Father will send in my name, he will teach you all things and bring to your remembrance all that I have said to you.

^16:7 I tell you the truth: it is to your advantage that I go away, for if I do not go away, the Helper will not come to you. But if I go, I will send him to you.

The Father speaks to the Son

In the story of the Baptism of Jesus, the Father speaks to the Son (Mark 1:11; cf. Luke 3:22):

And a voice came from heaven, “You are my beloved Son; with you I am well pleased.”

The Son communes with the Father

The Son prays to and communes with the Father in, for example, Matthew 11:25–26, Matthew 26:39, John 11:41, and John 12:27–28. Here's Matthew 11:

At that time Jesus declared, “I thank you, Father, Lord of heaven and earth, that you have hidden these things from the wise and understanding and revealed them to little children; ²⁶ yes, Father, for such was your gracious will.

The Holy Spirit prays to the Father

Finally, Berkhof refers to the Holy Spirit's intercession for believers in Romans 8:26:

Likewise the Spirit helps us in our weakness. For we do not know what to pray for as we ought, but the Spirit himself intercedes for us with groanings too deep for words.

This categorization and these passages come from Berkhof's Systematic Theology, 1.8.B.2.b. All quotations come from the English Standard Version of the Bible.