In the Levitical system (which we know is a precursor of the Refreshed Covenant), no one was allowed to enter the Most Holy Place in the Tabernacle/Temple, except for the High Priest, and then only once a year on the Day of Atonement to sprinkle blood on the Mercy Seat.
Fast forward to c. AD 33.
John gives us the most complete description of the empty tomb out of the four Gospels (since John is the Culmination/Man Gospel, this is to be expected). Mary Magdalene sees the empty tomb after Peter and John, but she sees something they did not.
And she saw two angels in white, sitting, one at the head and the other at the feet, where the body of Jesus had lain. (John 20:12)
Remind us of anything? It should.
And you shall make two cherubim of gold; of hammered work you shall make them at the two ends of the mercy seat. Make one cherub at one end, and the other cherub at the other end; you shall make the cherubim at the two ends of it with the mercy seat. And the cherubim shall stretch out their wings, and they shall face one another, the faces of the cherubim shall be toward the mercy seat. (Exodus 25:18-20)
The tomb became the new Most Holy Place. The veil of the Temple Most Holy Place was ripped, and three days later the veil of the tomb was ripped/rolled away. God descended OUT of the Most Holy Place to meet with us and bring salvation (this is very important in considerations of worship).
Who goes into this Most Holy Place?
Peter therefore went out, and the other disciple [John], and were going to the tomb. So they both ran together, and the other disciple outran Peter and came to the tomb first. And he, stooping down and looking in, saw the linen clothes lying there; yet he did not go in. Then Simon Peter came, following him, and went into the tomb, and he saw the linen clothes lying there, and the handkerchief that had been around His head, not lying with the linen clothes, but folded together in a place by itself. Then the other disciple, who came to the tomb first, went in also, and he saw and believed. (John 20:3-8)
Why does John stop and let Peter enter first?
I think it is because Peter represents the new High Priest (not the Great High Priest, mind you). John is not. Peter is the Rock on which the Church would be built. He enters the Most Holy Place, and sees the blood on the Mercy Seat. John sees that the veil has been torn, and enters after Peter, because there is no more separation between God and His people.
It was the greatest and final Day of Atonement.