The shorter Johannine lesson takes up two of John's most prominent agendas, namely, the relationship of sight to faith and the problem of elitist leaders in the church. In John, someone else is invariably closer to Jesus than Peter, and it is most often the anonymous "disciple whom Jesus loved." Here the two of them race to the tomb, and predictably, Peter not only loses the footrace, but though he sees first what is (or is not) inside the tomb, the other disciple believes first. John's point for elitists in the church seems to be that what makes a disciple is not a person's striving or priority, but Jesus' love.
Niedner, Frederick A. Jr., "The Ressurrection of our Lord Series A" (1989). Institute of Liturgical Studies Occasional Papers. Paper 34.