Today, in honoring St. John, the Beloved Disciple, we should note the passages where he is contrasted with St. Peter. First, at the Last Supper, when John was lying against Jesus, Peter instructed him to find out from Jesus who his betrayer would be.
They are contrasted again in today’s Gospel as they ran together to the tomb of Jesus. John ran faster than Peter, but then he waited to let Peter go in first.
Next we see them together when they breakfasted with Jesus by the Lake of Tiberius. Jesus takes Peter aside to give him charge over his sheep. Then, when Peter asked what John’s role would be, Jesus answered, “What if I wish him to remain until I come?”
In each of those instances John recognized Peter’s authority, while Peter recognized John’s deep relationship with Jesus.
The lesson for us there is that we must have respect for the authority of Church leaders, while they must have respect for good peoples' possibly closer relationship with the Lord.