The importance of establishing St. Peter's ministry in Rome may be boiled down to authority and more specifically the historic existence and continuance of the Office of Vicar held by St. Peter.
Many modern day academics enjoy setting St. Peter and St. Paul in enmity with one another; however, the over emphasis of Galatians 2:11-14 by modern scholarship fails to acknowledge that even though they had a disagreement their mission of spreading the Gospel was the same.
"Our apostles also knew, through our Lord Jesus Christ, that there would be strife on account of the office of the episcopate. For this reason, therefore... they appointed those [ministers] already mentioned, and afterwards gave instructions, that when these should fall asleep, other approved men should succeed them in their ministry."
"Great crowds, including even children too young to walk, assembled on the Mount of Olives and after suitable hymns, and antiphons, and readings, they returned in procession to Jerusalem, escorting the bishop, and bearing palms and branches of olives before him."