1) Both underwent a name change from Simon and Saul to Peter and Paul.
2) The change of names took place by entering into prayer: Simon entered into the prayer of Christ to the Father as in Luke 9, 18: “as He was praying in private, His disciples also were with Him;” Saul was recognized by Ananias in Acts 9, 12 “For behold, he is praying.” (Paul later remarked, "The time came when he who had set me apart before I was born and called my by his favor chose to reveal his Son to me, that I might spread among the Gentiles the good tidings concerning him" [Gal 1, 15]). As an act of self-gift, prayer produces an ontological change in the individual that actualizes him as person. Hence, as the ontology changes, the name changes.3) The Person of Christ as the Logos of the Father cannot be known by external sensation because "No one knows the Son except the Father; nor does anyone now the Father except the Son and him to whom the Son chooses to reveal him" (Mt 11, 27). Therefore, only those drawn by the Spirit of the Father in prayer can know the Person of the Son (Jn 6, 44). Since the Person of Christ is revealed to be prayer (see Luke 6, 12; 9, 18; 9, 28), when one begins to truly pray, one becomes “like” Christ and begins to experience being Christ from within. Cognizing (experiencing) the self as “like” Christ, one is then able to re-cognize the face of Christ outside of the self, and is enabled to say from within, “You are the Christ, the Son of the living God” (Mt. 16, 15). On this feast in 1991, John Paul II remarked, “a new bond between the human consciousness and the mystery of the living God” takes place. [For the development of this, see J. Ratzinger, “Behold the Pierced One” Ignatius (1986) 15-27].
4) Both Peter and Paul die as martyrs. John Paul also said, “The Church lives constantly through the inheritance of Peter which is the `ministry” (ministerium petrinum). She lives constantly through the inheritance of Paul which is the special charism of the proclamation of the Gospel. The Lord… stood by me and gave me strength to proclaim the word fully that all the Gentiles might hear it (2 Tm 4, 17). The double inheritance – ministerium petrinum and Pauline charism – leads us on today’s feast to Rome, to the place of both apostles’ birth into heaven, to fullness of life in God. This is the day on which the meaning of the keys of the kingdom of God is revealed n a special way. Whatever has been bound here on earth is bound in heaven – and whatever has been loosed on earth remains loosed in heaven (cf. Mt 16, 19): it has been sealed in the glory of the kingdom which never ends” (L’Osservatore Romano N. 27 (1198) – 8 July 1991, 1).
Appositely, St. Augustine comments, "Both apostles share the same feast day, for these two were one; and even though they suffered on different days, they were as one. Peter went first, and Paul followed. And so we celebrate this day made holy for us by the apostles' blood. Let us embrace what they believed, their life, their labors, their sufferings, their preaching and their confession of faith" (Sermon 295. 7-8).