The Gospel of Peter versus the Gospel of Matthew.

[RoadRunner79]

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Yes, it came later:

Date

The gospel is widely thought to date from after the composition of the four canonical gospels. Scholars are divided as to the exact date of the text, with Bart Ehrman placing it in the first half of the second century and considering it to have been compiled based on oral traditions about Jesus, independent of the canonical gospels.[5] The dating of the text depends to a certain extent on whether the text condemned by Serapion, Bishop of Antioch upon inspection at Rhossus is the same as the text that was discovered in modern times.[6] The Rhossus community had already been using it in their liturgy.[7]
John Dominic Crossan disagrees with most Biblical scholarship. Calling this gospel the "cross gospel", Crossan believes that this Gospel was written before the synoptic gospels of Mark, Matthew and Luke, and considers Peter to be a forerunner to those gospels.[8] Crossan's view is not accepted by other Biblical scholars.[9]
Later Western references, which condemn the work, such as Jerome and Decretum Gelasianum, traditionally connected to Pope Gelasius I, are apparently based upon the judgment of Eusebius, not upon a direct knowledge of the text.[10]

Wikipedia -- Gospel of Peter date.

But, not that much later:

Setting and date

The majority view among scholars is that Matthew was a product of the last quarter of the 1st century.[23][Notes 1] This makes it a work of the second generation of Christians, for whom the defining event was the destruction of Jerusalem and the Temple by the Romans in AD 70 in the course of the First Jewish–Roman War (AD 66–73); from this point on, what had begun with Jesus of Nazareth as a Jewish messianic movement became an increasingly Gentile phenomenon evolving in time into a separate religion.[24] The Christian community to which Matthew belonged, like many 1st-century Christians, was still part of the larger Jewish community: hence the designation Jewish Christian to describe them.[25] The relationship of Matthew to this wider world of Judaism remains a subject of study and contention, the principal question being to what extent, if any, Matthew's community had cut itself off from its Jewish roots.[26] Certainly there was conflict between Matthew's group and other Jewish groups, and it is generally agreed that the root of the conflict was the Matthew community's belief in Jesus as the Messiah and authoritative interpreter of the law, as one risen from the dead and uniquely endowed with divine authority.[27]
The author of Matthew wrote for a community of Greek-speaking Jewish Christians located probably in Syria (Antioch, the largest city in Roman Syria and the third-largest in the empire, is often mentioned).[28] Unlike Mark, Matthew never bothers to explain Jewish customs, since his intended audience was a Jewish one; unlike Luke, who traces Jesus' ancestry back to Adam, father of the human race, he traces it only to Abraham, father of the Jews; of his three presumed sources only "M", the material from his own community, refers to a "church" (ecclesia), an organised group with rules for keeping order; and the content of "M" suggests that this community was strict in keeping the Jewish law, holding that they must exceed the scribes and the Pharisees in "righteousness" (adherence to Jewish law).[29] Writing from within a Jewish-Christian community growing increasingly distant from other Jews and becoming increasingly Gentile in its membership and outlook, Matthew put down in his gospel his vision "of an assembly or church in which both Jew and Gentile would flourish together".[30]

Wikipedia -- Gospel of Matthew date

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And, so, why believe Matthew's account over that of Peter's?

Because it's not just about the age of the material.   The early Church disregarded  a number of materials, that where late (did not have the connection to the apostles), because they conflicted with the Gospel they where given.  Many of these Churches where said to be founded by the apostles and disciples within 100 years of the time that you are talking about.   The Gospel of Peter, if what we have today resembles what it was then (remember it was lost for about 1000 years), actually isn't that much different.  In corroborates a number of things in the orthodox Gospels.  There are reports that it was read in some churches, and there are also reports from those saying not to.  It has docetic overtones which where against the teaching of the bible.  It's not just the age, but also the history behind it.