What are Petrine Christianity and Pauline Christianity and how do they differ?

Answer 1

Pauline Christianity is a term used to refer a branch of Early Christianity associated with the beliefs and doctrines espoused by Paul the Apostle through his writings. Most of mainstream Christianity relies heavily on these teachings and considers them to be amplifications and explanations of the teachings of Jesus. Others perceive in Paul's writings teachings that are radically different from the original teachings of Jesus documented in the canonical gospels, early Acts and the rest of the New Testament, such as the Epistle of James. Pauline Christianity; Wikipedia

On Petrine Christianity, maybe you are referring to Petrine Primacy?

A number of Christian denominations and scholars hold that Simon Peter was the most prominent of the apostles, favored by Jesus of Nazareth with the first place of honor and authority.[1] This doctrine is known as the Primacy of Simon Peter or the Petrine Primacy (from the Latin Petrus=Peter). A number of traditions, most notably Roman Catholic, hold that Simon Peter, also known as Saint Peter or Cephas, was the first Bishop of Rome and was a martyr during the persecution of the emperor Nero. The early Christian church at this time, however, did not have precisely delineated functions for bishops or other official roles.[2].. Primacy of Peter; Wikipedia