...the reason maybe that Luke and Matthew give a different version is that they are talking about Jesus preaching on two occasions, a year apart.
A closer focus on this point: this was often an argument advanced by the Watchtower when I was associating with the Witnesses during the 1980s. It hasn't much critical scholarship to stand on.
Matthew's gospel appears to be a Jewish liturgical account, meant to be read in a synagogue of Jewish Christian worshipers who were Torah-observant. Along with their Torah readings on the Sabbath (Friday night or Saturday morning), they would read from Matthew's text. They saw no alteration to their Judaism in accepting Jesus of Nazareth as Messiah, neither did they change their Jewish practice.--See Acts 21:20ff.
Luke's gospel was once a single volume with the Acts of the Apostles. It was an evagelism tool, read in Christian meetings that took place once the Sabbath had concluded, usually on Saturday night (as Jews could not assemble with Gentile believers or assemble at all outside of synagogues or Temple according to the Law as long as it was Sabbath--and Sabbath ended on sundown Saturday). This might explain Paul's discourse on "the first day of the week," why it occured late at night, and why it lead to the death and subseqent resurrection of Eutychus. (See Acts 20:7-12) This text of Luke's became used for the new "Lord's Day" services as liturgy of a new type, meetings which were held "on the first day of the week," the day after the Jewish Sabbath, or our Saturday night or Sunday mornings. Eventually the two services would merge on Sunday once Christianity became Rome's state religion.
At the canonization of the New Testament, the gospels were separated into their own section, and Luke was split from its second half with the gospel of John dividing it from Acts.
The discourse in which the Lord's Prayer occurs in both Matthew and Luke is the exact same event. The difference is that Matthew records it as Jewish liturgy and Luke records it as Christian evagelism to the Gentiles.
As Jewish liturgy, Jesus is the New Moses to whom his disciples "climb the New Sinai" to hear the "Sermon on the Mount." From there they hear "the Torah" explained, receiving the Beatitudes much like Israel received the Ten Commandments: "Blessed are the meek...Blessed are the..." etc. This Jesus in Matthew claims he comes not to destroy the Law but to fulfill it and curses those who teach against it. He then teaches his followers how to pray. The prayer is not unique, however, but very much in line with common Jewish themes along the line of rabbincal teachers of Jesus' day.
Luke does not compose the event as does Matthew. He does not draw the event as if the people are all Jews. He writes for an international audience, in a way that the reader can see themselves as part of those approaching Jesus. Instead of a mountain, Jesus is on a plain and approachable. Instead of a figure like Moses, Jesus is very human. He does not speak of the Mosaic Law. Instead of fulling the Law, Jesus is seen as comming to redeem and teach the world, not merely the Jews. Luke is foreshadowing the events that will lead to Paul's ministry in the Acts of the Apostles.
Pope Francis' discussion about the Our Father is a little skewed in the press because most reporters don't know the history of where Christianity got the prayer in the first place. In fact, Christianity prayed the Our Father/Lord's Prayer first before the Gospels were written. It was an oral instruction and part of memorized instruction among Christians. Only later was it inserted into Christian Scripture. Most of the gospels we have come from "saying sources" like M and Q and L. Q, in fact, may have been the Aramaic sayings source that Papias identified as "Matthew's gospel," in theory.
All nomative academia agrees that none of the narrative in the gospels comes from the original authors but was added much later to these sayings that the Christians attributed to what they claimed to be oral teachings attributed to memory from Jesus of Nazareth. This includes the Lord's Prayer which had been recited by this time by memory in a liturgical setting prior to the Communion service in Christian services before anyone had written it as gospel.