In today’s Gospel 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 took 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' closer relationship with the Lord.