Why? Christ clearly stated it to Pater. When he first saw Simon, "Jesus looked at him, and said, So you are Simon the son of John? You shall be called Cephas (which means Peter)" (John 1:42). The word Cephas is merely the transliteration of the Aramaic Kepha into Greek. Later, after Peter and the other disciples had been with Christ for some time, they went to Caesarea Philippi, where Peter made his profession of faith: "You are the Christ, the Son of the living God" (Matt. 16:16). Jesus told him that this truth was specially revealed to him, and then he solemnly reiterated: "And I tell you, you are Peter" (Matt. 16:18). To this was added the promise that the Church would be founded, in some way, on Peter (Matt. 16:18).
Then two important things were told the apostle. "Whatever you bind on earth shall be bound in heaven, and whatever you loose on earth shall be loosed in heaven" (Matt. 16:19). Here Peter was singled out for the authority that provides for the forgiveness of sins and the making of disciplinary rules. Later the apostles as a whole would be given similar power [Matt.18:18], but here Peter received it in a special sense.
Peter alone was promised something else also: "I will give you the keys of the kingdom of heaven" (Matt. 16:19). In ancient times, keys were the hallmark of authority. A walled city might have one great gate; and that gate had one great lock, worked by one great key. To be given the key to the cityan honor that exists even today, though its import is lostmeant to be given free access to and authority over the city. The city to which Peter was given the keys was the heavenly city itself. This symbolism for authority is used elsewhere in the Bible (Is. 22:22, Rev. 1:18).
Says you. This does not mean the same thing was not told or given to the other twelve apostles. It just wasn't recorded. Twelve walls, twelve gates, twelve apostles.
We should all keep in mind too that the fact that Peter was a repentent sinner really has no bearing whatsoever on the matter of the "gates of hell" prevailing or not prevailing. God has consistently chosen to use sinful human beings to infallibly teach "faith and morals" to other sinful human beings, even before He began to build His Church.
For example, most Christians agree that every single "Book" in the Bible infallibly teaches "faith and morals", even though each one was written by a sinful human being, under the inerrant and unceasing guidance of the Holy Spirit.
David wrote many of the Psalms in the Old Testament after he committed adultery and murder, Saul/Paul wrote his New Testament letters after he persecuted Jesus, and Peter wrote 1 Peter and 2 Peter as now contained in the New Testament after he publicly denied his Lord.
In His perfect and sovereign will, God has also chosen to have a series of sinful human beings appointed to be His Pope (serving as the "Prime Minister" to the King of Kings), starting with Peter, and has given them a number of special gifts, including the certainty that they will infallibly teach "faith and morals according to Almighty God", both to the Church that Jesus built (as He had solemnly promised he would), as well as to the rest of the world. This infallibility when teaching "faith and morals" extends also to the "Magisterium", or "Teaching Authority" of the Church that Jesus built (as He had solemnly promised), and consists of the Pope and the Bishops in union with the Pope in their teaching office or authority.
With God, all things are possible