What is the difference between Christ’s anointing and the anointing of the Chrism (or Myron) of those who are newly baptized?

The Spirit descended and remained upon the Word of God, who was incarnate, being united with Him and equal to Him. Hence, we the believers have received the right to receive the Holy Spirit as a free gift.

St. Ambrose says, “There was therefore a pouring out upon us of the Spirit, but upon the Lord Jesus, when He was in the form of man, the Spirit abode … Around us is the liberality of the Giver in abundant provision, in Him abides forever the fullness of the Spirit. He shed forth then what He deemed to be sufficient for us, and what was shed forth is not separated nor divided; but He has a unity of fullness wherewith He may enlighten the sight of our hearts

1411 St. Athanasius of Alexandria, Four Discourses Against the Arians 1.12.47 (NPNF II/4:334).

1412 St. Athanasius of Alexandria, Four Discourses Against the Arians 1.12.50 (NPNF II/4:336).

1413 St. Ambrose of Milan, On the Mysteries 7.42 (NPNF II/10:322).

according to what our strength is capable of. Lastly, we receive so much as the advancing of our mind acquires, for the fullness of the grace of the Spirit is indivisible, but is shared in by us according to the capacity of our own nature.”¹⁴¹⁴

He also says, “David has reason to cry, like a man renewed, ‘And I will go in to the altar of my God, to God, who gives joy to my youth’ (Ps. 43:4). Even as he had said earlier that he had grown old among his enemies, as we read in the sixth psalm, here he says that youth has been restored to him after the lengthy old age of man’s fall. For we are renewed through the regeneration experienced in baptism, we are renewed through the outpouring of the Holy Spirit, we shall be renewed also through the resurrection, even as he says in a later passage, ‘your youth is renewed like the eagle’s’ (Ps. 103:5). Learn the manner of our renewal: ‘You shall sprinkle me with hyssop and I shall be cleansed, you shall wash me and 1 shall be made whiter than snow’ (Ps. 51:9). And in Isaiah, ‘Though your sins are like scarlet, they shall be as white as snow’ (Isa. 1:18). One who changes from the darkness of sin into the light of virtue and into grace is properly renewed. Thus, one who earlier was stained with foul defilement may shine with a brightness that is whiter than snow.”¹⁴¹⁵