Jesus is said to be born is Bethlehem, But why does Pope stay in Rome (Vatican)?  

I can see why you have a "3" in that little star of yours. Do you make it a point to insult Catholics every time you see a discussion mentioning them? That "old man" is my Holy Father, and I'm not about to let you slide with insulting him like that.

As for why the Vatican is in Rome:

http://www.ewtn.com/faith/teachings/papaa1.htm

With his vigorous outspokenness, Peter inevitably came into conflict with the Jewish authorities, and twice the high priests had him arrested. We are told that he was miraculously freed of his prison chains, and astonished the other Apostles by suddenly appearing back among them. Peter now preached in the seaports of Joppa and Lydda, where he met men of many races, and in Caesarea, where he converted the first Gentile, a man named Cornelius. Realizing that the sect must win its greatest support from Gentiles, Peter helped to shape the early policy towards them. Its growing eminence led to his election as bishop of the see of Antioch. How long he remained there, or how or when he came to Rome, we do not know. The evidence seems to establish the fact that his last years were spent in Rome as bishop. The belief that he suffered martyrdom there during the reign of Nero in the same year as St. Paul is soundly based on the writings of three early Fathers, St. Irenaeus, Clement of Alexandria, and Tertullian.[1]

Jesus Himself told Peter "I tell you that you are Peter, and upon this rock, I will build my church." Guess where St. Peter's Basilica is located? Right above St. Peter's tomb.

The latest archeological findings indicate that St. Peter's Church in Rome rises over the site of his tomb, as Pius XII announced at the close of the Holy Year of 1950.