Short answer, the milk:
The Spirit of God is the Father's consciousness that He shares perfectly with His begotten Son, who then shares his spirit (consciousness) with us. In this way, the Spirit of God can only be accessed through the Son.
The longer more complete answer, the meat:
There is only one eternal Spirit and this would be the Holy Spirit of God. And by God, I'm referring to "the only true God" of Jesus Christ.
And this is life eternal, that they might know thee the only true God, and Jesus Christ, whom thou hast sent
Jesus tells the woman at the well that his God and Father is a Spirit.
23 But the hour cometh, and now is, when the true worshippers shall worship the Father in spirit and in truth: for the Father seeketh such to worship him. 24 God is a Spirit: and they that worship him must worship him in spirit and in truth.
Paul makes it clear that there is one Spirit.
4 There is one body, and one Spirit, even as ye are called in one hope of your calling; 5 One Lord, one faith, one baptism, 6 One God and Father of all, who is above all, and through all, and in you all.
Therefore, I will be continuing this explanation with the premise that the Holy Spirit of God is our Father's Spirit and not a completely other eternal spirit and/or person.
To use the parlance of the times, The Holy Spirit of God is the Father's consciousness.
Humans, born in the image of the Creator, also have a consciousness. This is our individualized soul/spirit. It's who each of us identifies as; what you are besides your flesh suit.
Modern man-made concepts concerning consciousness are still undeveloped and not fully defined. To put it simply your consciousness is your instantaneous thoughts and emotions.
The Father shares His Spirit (consciousness) fully and perfectly with the Son. That is how we say the Son is "in" the Father and vice-versa. They are one.
Our Master, the Son, likewise shares his holy spirit (his consciousness) with us. That is how we are one with the Father. We are in our Master who is perfectly in his God.
Therefore being by the right hand of God exalted, and having received of the Father the promise of the holy spirit, he hath shed forth this, which ye now see and hear.
The holy spirit mentioned here is part of the glory promise Jesus received when he was exalted as we read above; a counterpart to his new spirit body. That is how we know this holy spirit is something different than the Spirit of God which is the Father's Spirit. He was immersed with the Father's Spirit during his baptism. Still one Spirit, however, because they share consciousness.
In the end, all of the children of Elohim will inherit a holy spirit, which will be one with the Spirit of the Father. At that time, God will be all in all. There is only one eternal consciousness and this is the Spirit of YHWH, the Eternal One.
Who is the Parakletos
But the Parakletos, [G3875] this holy spirit, whom the Father will send in my name, he shall teach you all things, and bring all things to your remembrance, whatsoever I have said unto you.
But when the Parakletos [G3875] is come, whom I will send unto you from the Father, even this spirit of truth, which proceedeth from the Father, he shall testify of me:
Nevertheless I tell you the truth; It is expedient for you that I go away: for if I go not away, the Parakletos [G3875] will not come unto you; but if I depart, I will send him unto you.
The parakletos in chapter 14 is the Risen Son. I can say this with confidence because John, the same author of the verse in question, also wrote this verse:
1 John 2:1
My little children, these things write I unto you, that ye sin not. And if any man sin, we have a Parakletos with the Father, Jesus Christ the righteous:
It's important to note that in the 14th chapter of John, Jesus has not been glorified. He is speaking as the human Messiah, not the Risen Son in all his glory of his inheritance. The Risen Son inherits a spirit body and a spirit consciousness that he can now share just as his God and Father shared His Spirit (consciousness) perfectly with him.
The Risen Son is the Vine and we are the branches. The Spirit of God proceeds from the Father and moves through the Son to the rest of the children of God.
The Vine is the only pathway to the Father, and therefore, is the only doorway the Father uses to share with us His Spirit; upgrading our corrupted old spirit seed with the new incorruptible seed by way of the Risen Son who is the seed inside all of us who are born again.
In conclusion, the Father sends His Holy Spirit through the Son and in his name. You can't receive it any other way. Jesus is the way.