Well, there’s not too much of a difference as regards the primacy of Peter, and a plain reading of Scripture will show you how Peter was singled out, but mainly I would say the issue is political between the two.
I commend you on finding the right Church.
Here’s something I found:
On the basis of Mark 3:16, 9:2, Luke 24:34 and 1 Corinthians 15:5, the Catechism of the Catholic Church describes Peter as holding first place among the apostles. It speaks of Peter as the rock on which, because of Peter’s faith, Christ said in Matthew 16:18 he would build his Church, which he declared would be victorious over the powers of death. In Luke 22:32, Jesus gave Peter the mission to keep his faith after every lapse and to strengthen his brothers in it. The Catechism of the Catholic Church sees the power of the keys that Jesus promised to Peter alone in Matthew 16:19 as signifying authority to govern the house of God, that is, the Church, an authority that Jesus after his resurrection confirmed for Peter by instructing him in John 21:15–17 to feed Christ’s sheep. The power to bind and loose, conferred on all the apostles jointly and to Peter in particular (Matthew 16:19) is seen in the Catechism of the Catholic Church as authority to absolve sins, to pronounce judgments on doctrine and to make decisions on Church discipline.
Many of the Church Fathers spoke of ecumenical councils and the Bishop of Rome as possessing a reliable authority to teach the content of scripture and tradition.
Pope St. Clement of Rome, c. 99, stated in a letter to the Corinthians: "Indeed you will give joy and gladness to us, if having become obedient to what we have written through the Holy Spirit, you will cut out the unlawful application of your zeal according to the exhortation which we have made in this epistle concerning peace and union."
St. Clement of Alexandria wrote on the primacy of Peter c. 200: “the blessed Peter, the chosen, the pre-eminent, the first among the disciples, for whom alone with Himself the Savior paid the tribute…”
Also of note is the very early Easter dating crisis which hit the early Church between what liturgical calender to use, which was settled by the Pope.