How is Christian regeneration through the Spirit different from the birth of Jesus?

Christians, those genuinely born of the Spirit (John 3:1-7), are different than Jesus is. In John 8:58, Jesus stated, "... Truly, truly, I say to you, before Abraham was, I am" (cf. Exod. 3:13-14). So, Christ's existence is eternal, a Christian's existence isn't. Moreover, Christ has all the attributes of God, Christians don't. See "What are the Attributes of God?" below.

Christ is very God of very God, Christians aren't. As the Nicene Creed (325 A.D.) states:

And in one Lord Jesus Christ, the only-begotten Son of God, begotten of the Father before all worlds; God of God, Light of Light, very God of very God; begotten, not made, being of one substance with the Father, by whom all things were made.

Moreover, as can be seen above, Jesus is the only-begotten Son of God (John 3:16), but Christians are only the adopted sons or daughters of God (Rom. 8:14-17; Gal. 4:4-8). While an adopted son or daughter has many of the privileges of sonship, they are not God incarnate; the God-man (John 1:14; cf. John 1:1, 18).

Christ is eternal - there was never a time He wasn't God. In the fullness of time, Christ took to Himself a human body (cf. Matt. 1:21-23; Luke 2:7; Gal. 4:4). The Chalcedon Creed (451 A.D.) summarizes the teaching of the incarnation of Christ pointing out these truths:

1. Jesus has two natures - He is God and man.
2. Each nature is full and complete - He is fully God and fully man.
3. Each nature is and remains distinct.
4. Christ is only one Person.
5. Things that are true of only one nature are nonetheless true of the Person of Christ.

As can be clearly seen above, Jesus is 100% God and 100% man, as compared to the Christian who is only 100% human. The absolute fullness of these five truths of the Chalcedon Creed can't be applied to a mere man, but only to Christ alone. Indeed, though a Christian's re-birth is through the Spirit (John 3:1-7) this does not make him God; it simply restores him to the people made in God's image (cf. Gen. 1:26-28). It reveals that they are elect, but still sinners (1 John 1:8-10); but Christ never sinned (2 Cor. 5:21; Heb. 4:15; 1 Pet. 2:22).

There are only three members of the Godhead; not God the Father, God the Son, God the Holy Spirit, and all the number of the elect. For election, see "Calvinism and Matthew 13?" below. While Jesus' human nature is not part of the Trinity, His divine nature is. Just because a Christian is born of the Spirit and the Spirit of God dwells in them (2 Tim. 1:14), doesn't mean they merge with the Spirit and become Gods. The (1647) Westminster Confession of Faith 2.3 states:

In the unity of the Godhead there be three Persons of one substance, power, and eternity: God the Father, God the Son, and God the Holy Ghost. The Father is of none, neither begotten nor proceeding; the Son is eternally begotten of the Father; the Holy Ghost eternally proceeding from the Father and the Son.

I hope this helps.

Related Links:

What are the Attributes of God?
Calvinism and Matthew 13?
The Holy Spirit in the Westminster Standards?
The Articles of Faith