What role does the Holy Ghost fill that the Father can't?

If we are seeking a possible historical source that gave rise to the prominent position of the "Holy Ghost" in the Trinity dogma then I would look to the Egyptian deity Horus (aka Her, Heru, Hor, Har) who was the lastborn of the first five original gods.

The reason I point to Horus is that he is a bird:

https://youtu.be/JXKsyvTDMEs

The significance of Horus being a bird is that it means that he is mobile.

In practical terms it meant that Horus could be on hand to effect divine will in distant places. It also means that he can enter into various persons.

While the Jewish and Christian symbolism is often that of a dove, such as with Noah and Jesus' baptism, the more common Jewish and Christian symbolism for this kind of "mobility" is that of "breath" and other forms of moving air.

Now the Son of God, Jesus, is essentially God's "body" or "face" (like the Executive branch of the US government), the "breath" is the invisible but remotely and inwardly acting agent in the divine economy. So God expresses himself through the Messiah but the Messiah operates by the indwelling and outbreathed effects of the divine breath aka the PNEUMA.

This, I believe, properly explains the conception and historic rationale of the concept of the PNEUMA and "the Holy Breath" aka "Holy Spirit".

Tertullian was the first to use the word "Trinity"/"Triad". He was not referring to the Godhead being three equal divine Persons but rather he was referring to the structure of God's "economy". The "economy" was the way that God expressed himself and handled his dealings, which Tertullian conceived of as a triad. For him, God handled his dealing through a triad of which he Himself was the founding member.

Limiting God

Is it hard to reduce God to a checklist

"But will God indeed dwell on the earth? Behold, heaven and the highest heaven cannot contain You, how much less this house which I have built!" 1 Kings 8:27

What God cannot do

Remember the Angel's words

"For nothing will be impossible with God.” Luke 1:37

So asking what is this that God cannot do may be a hard question.

Old Testament

In the OT the Holy Ghost is mentioned several times with characteristics and activities that may overlap with those of the Father, but it is mentioned separately in the Scriptures as an entity (similar as the Father is mentioned).

"Create in me a clean heart, O God, And renew a steadfast spirit within me. Do not cast me away from Your presence And do not take Your Holy Spirit from me." Psalm 51:10,11

New Testament

In the NT Jesus explains the role of the Holy Ghost as comforter, guide, teacher.

Jesus portrays the Father deciding and accompanying in the passing of every bird

"Are not two sparrows sold for a cent? And yet not one of them will fall to the ground apart from your Father." Matthew 10:29

A humble task for the Father.

In John 14 Jesus says

"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 said to you." John 14:26

So, in words of the Son, the Spirit is sent by the Father.

In John 15 Jesus addresses again the issue

"When the Helper comes, whom I will send to you from the Father, that is the Spirit of truth who proceeds from the Father, He will testify about Me" John 15:26

Since the idea of the Father sends the Spirit is repeated for the 2nd time in the same conversation, it may be an important one. I understand that the Father and the Spirit are two persons.

The Apostles say the Spirit is an intercessor, comforter and guide.

Paul in Romans 8 says

"In the same way the Spirit also helps our weakness; for we do not know how to pray as we should, but the Spirit Himself intercedes for us with groanings too deep for words" Rom 8:26

It is given that intercedes for us upon the Father, so there are different roles again.

Then he tells us that Jesus is interceding

"who is the one who condemns? Christ Jesus is He who died, yes, rather who was raised, who is at the right hand of God, who also intercedes for us." Rom 8:34

Paul says that Jesus is the intercessor, so there are some overlapping roles not only between the Father and the Spirit, but also between the Spirit and the Son.

The mystery of the Trinity is hard to explain for us mortals, let the Spirit guide us to the truth.

The question seems to assume a degree of distinction that does not exist. If it can be allowed that scripture draws a distinction between God and the Word of God:

and the Word was with God

and if that distinction can be allowed to be both clear and yet not representative of total ontological separation:

and the Word was God

then the same can be understood regarding the Spirit of God.

As Geremia has pointed out, both the Word (Son) of God and the Spirit of God are sent and Aquinas has correctly reasoned that, as this action requires a sending agency of greater authority and that the sender is God of whom there is no greater authority, the primary distinction in role lies just here: While both the Spirit and the Word always accomplish perfectly and completely what God intends because they are both eternally God, they nonetheless operate under the authority of the sender. This is not a distinction in 'being' but in 'function' and the purpose is that we might know God personally and intimately as Father.

The Father represents a similar distinction from God because GOD is not, by default, a Father to all (1 John 2:23). GOD works by the power of His Spirit to transform slaves of sin into children for Himself in and through His Son:

But when the fullness of time had come, God sent forth his Son, born of woman, born under the law, to redeem those who were under the law, so that we might receive adoption as sons. And because you are sons, God has sent the Spirit of his Son into our hearts, crying, “Abba! Father!” So you are no longer a slave, but a son, and if a son, then an heir through God. - Galatians 4:4-7

If anyone is in Christ he is a new creation. The grace of God has granted that, by an appeal of faith to the sacrifice of the Son of God for reconciliation, the Holy Spirit of God indwells the believer's heart to promote conformity to the life of the Son whom He loves and by that Spirit we cry "Abba, Father".

The new birth (John 3) is entirely Trinitarian. God becomes your Father when you accept the revelation of His Son and His Spirit inaugurates and guarantees Christ in you, the hope of glory.

No one who denies the Son has the Father. Whoever confesses the Son has the Father also. - 1 John 2:23

Jesus clearly identified the distinct role of the Holy Spirit in John 16:

⁵But now I go my way to him that sent me; and none of you asketh me, Whither goest thou? ⁶But because I have said these things unto you, sorrow hath filled your heart. ⁷Nevertheless I tell you the truth; It is expedient for you that I go away: for if I go not away, the Comforter will not come unto you; but if I depart, I will send him unto you.
⁸And when he is come, he will reprove the world of sin, and of righteousness, and of judgment:
⁹Of sin, because they believe not on me;
¹⁰Of righteousness, because I go to my Father, and ye see me no more;
¹¹Of judgment, because the prince of this world is judged.

So, it is the distinct role of the Holy Spirit to reprove the world of:

• SIN - not transgression of the Law of Moses, but specifically unbelief that Jesus is God.

  In John 8:23-24, Jesus reproved the Jews concerning belief in him:

  ²³... Ye are from beneath; I am from above: ye are of this world; I am not of this world. ²⁴I said therefore unto you, that ye shall die in your sins: for if ye believe not that I am he, ye shall die in your sins.

  When Jesus said to the Jews, "if ye believe not that I am he", it is important to understand who the "he" is referring to. It can only be a reference to Jesus' prior statement in verse 19:

  ¹⁹... Ye neither know me, nor my Father: if ye had known me, ye should have known my Father also.

• RIGHTEOUSNESS - not that which is associated with the Law of Moses, but specifically that which is by faith of Jesus Christ (Romans 3:20-22 KJV):

  ²⁰Therefore by the deeds of the law there shall no flesh be justified in his sight: for by the law is the knowledge of sin. ²¹But now the righteousness of God without the law is manifested, being witnessed by the law and the prophets; ²²Even the righteousness of God which is by faith of Jesus Christ unto all and upon all them that believe: for there is no difference:

• JUDGEMENT - not judgement that is associated with evil (harm, distress, disaster, etc) that comes in this life as a consequence of transgressing the Law of Moses, but specifically that associated with the evil one, i.e. "the prince of this world". He has been judged and all those whose father he is are judged along with him (John 8:44-47 KJV):

  ⁴⁴Ye are of your father the devil, and the lusts of your father ye will do. He was a murderer from the beginning, and abode not in the truth, because there is no truth in him. When he speaketh a lie, he speaketh of his own: for he is a liar, and the father of it. ⁴⁵And because I tell you the truth, ye believe me not.
  ⁴⁶Which of you convinceth me of sin? And if I say the truth, why do ye not believe me? ⁴⁷He that is of God heareth God's words: ye therefore hear them not, because ye are not of God.

The Father sent the Son, and while he walked in 1st century Palestine he reproved the Jews in regard to sin, righteousness and judgement, but when the Son returned in glory to the Father, it was given to the Holy Spirit to then reprove the world in the same regard until Jesus returns.

A problem when trying to grapple with questions like this is the way our word ‘persons’ is not really what the original language meaning was. But I don’t want to delve into that as you are specifically asking for any distinct roles the Holy Spirit has that the Father has not. Well, could this be a case of, not ‘cannot have’ but ‘will not have’? Further, is the query as to “what makes the Holy Ghost different enough in trinitarian doctrine to make him a separate [sic] person” back to front? If it were put the other way around, would an answer be forthcoming? Consider: “What makes the distinct role of the Holy Spirit in the Trinity sufficient to clarify his person-hood as distinct from the person-hood of the Father (and the Son)?” Yet I must stick to YOUR question, and not try to inflict my one on you! However, if you bear in mind the importance of getting the right trinitarian order established, then the answers are – although never easy – a bit easier to fathom.

There is a patristic maxim that may well apply here: opera ad extra trinitatis indivisa sunt – meaning, “The external works of the Trinity are undivided” – even if these opera are ‘appropriated’ to one or other person of the Godhead. Let me now quote an example of that point in the book below:

“The resurrection [of Christ] while chiefly attributed to the Father (Acts 2:32; 17:31; Rom. 8:11; 1 Cor. 15:15), is also seen to be an action of the Son (Jn. 2:19-21; 10:17-18). But, according to Paul, it is also ‘through the Spirit of holiness’ that Jesus ‘was declared with power to be the Son of God, by his resurrection from the dead’ (Rom. 1:4). This, coupled with the ‘faithful saying’ that Jesus was ‘vindicated by the Spirit’ (a reference to the resurrection as his ‘justification’, 1 Tim. 3:16), and Peter’s words ‘He was put to death in the body but made alive by the Spirit…’ (1 Pet. 3:18), underlines the Spirit’s role in the resurrection.” Sinclair Ferguson, The Holy Spirit, p53 (ivp 1996)

I would suggest that the above example obtains with every work of the Holy Spirit you care to mention, notwithstanding Geremia’s answer, for that deals with the manner of ‘sending’ the Spirit into the world for His mission, and is not actually dealing with the work of the Spirit. The logistics of organising a mission cannot be the same as the actual work of mission done by the one sent on it. The Father is never spoken of in scripture as “being sent”, but of sending. That, alone, signifies one predominant aspect of the Father, but the Son also sends, and “the Spirit gives life” – 2 Corinthians 3:6 – and gives to each one certain of His gifts. To ‘give’ in those senses is equal to ‘sending’ life and gifts to believers. Yet there is a clear emphasis in scripture of the extent of the three persons’ sending.

Perhaps the degree of scriptural references to the various functions and roles of the three persons in the Godhead shows both their distinctions and their oneness. There is order within the Godhead. The Son relates to the Father in a particular way, and the Holy Spirit relates to the Father and the Son in particular ways. This shows that the Son is not the Father, and the Holy Spirit is neither the Father nor the Son. Yet their oneness is indisputable, for they all have the one Being of God.

That is why ‘opera ad extra trinitatis indivisa sunt’ – “The external works of the Trinity are undivided” – applies to why the Holy Spirit does a particular work, showing him to be distinct as one of the persons of the Trinity, yet the Triune God remaining utterly one in the three persons. This is no different to how the Son does a particular work, showing him to be distinct as one of the persons of the Trinity, yet the Triune God remaining utterly one in the three persons.