Three of the writers of the Gospels—Mark, Luke and John—use the term "Kingdom of God" to refer to the Kingdom by name.
"Kingdom of heaven" is a term used exclusively by Matthew, with 32 references in his account of the life of Jesus Christ. However, he uses the terms "kingdom of God" and "kingdom of heaven" interchangeably. In Matthew 19:23-24 Matthew 19:23-24  Then said Jesus to his disciples, Truly I say to you, That a rich man shall hardly enter into the kingdom of heaven.
 And again I say to you, It is easier for a camel to go through the eye of a needle, than for a rich man to enter into the kingdom of God.
American King James Version×, he uses the terms in consecutive verses, clearly implying that they were synonymous. Often he calls it simply "the kingdom."
Why did Matthew call it "the kingdom of heaven"? Because he was writing primarily to a Jewish audience. According to the Jewish Encyclopedia article "kingdom of God," Malkut Shamayim, meaning "Kingdom of heaven," is the way the Jews of that day designated the Kingdom of God. They understood it in two ways. One was the literal Kingdom as prophesied in Daniel 2, where this Kingdom is represented by a stone "from heaven" that comes down and smashes the kingdoms of this world and then grows to fill the whole earth. And indeed, heaven is the place from which Jesus will come to establish the Kingdom on earth. In short, it is the Kingdom of heaven because it is the Kingdom from heaven (not because, as some now contend, it is a kingdom in heaven).
Another aspect of Jewish understanding of the Kingdom of God in Christ's day concerned the "reign or sovereignty of God as contrasted with the kingdom of worldly powers" (ibid.). In other words, when one committed himself to total obedience to God's laws and His revealed way of life, he was submitting himself to the authority of the "Kingdom of God" or "Kingdom of heaven" in contrast to all other human or worldly authority. At times and in certain contexts Jesus appeared to use the term with this meaning.
Another factor, as is pointed out in some commentaries, is that because of the reluctance the Jews had about pronouncing or using God's name, Matthew substituted "kingdom of heaven" for "kingdom of God." This is apparent when we compare such passages as Matthew 4:17 Matthew 4:17From that time Jesus began to preach, and to say, Repent: for the kingdom of heaven is at hand.
American King James Version×with Mark 1:15 Mark 1:15And saying, The time is fulfilled, and the kingdom of God is at hand: repent you, and believe the gospel.
American King James Version×and Matthew 5:3 Matthew 5:3Blessed are the poor in spirit: for theirs is the kingdom of heaven.
American King James Version×with Luke 6:20 Luke 6:20And he lifted up his eyes on his disciples, and said, Blessed be you poor: for yours is the kingdom of God.
American King James Version×, where Matthew uses the phrase "kingdom of heaven" while Mark and Luke use "kingdom of God" in quoting Jesus regarding the same events. It should be realized, however, that the Jews sometimes did use the term "kingdom of God" or "kingdom of the Lord" and not just "kingdom of heaven." Evidently, Jesus Himself used the terms interchangeably.
The apostle Paul usually uses the term "kingdom of God" in his letters. However, acknowledging the role of Jesus Christ as the Ruler of that Kingdom and the way by which we enter that Kingdom, he also calls it "the kingdom of Christ and God" (Ephesians 5:5 Ephesians 5:5For this you know, that no fornicator, nor unclean person, nor covetous man, who is an idolater, has any inheritance in the kingdom of Christ and of God.
American King James Version×). He also expresses the deep, loving relationship between God the Father and Jesus Christ by calling it "the kingdom of the Son of His love" (Colossians 1:13 Colossians 1:13 Who has delivered us from the power of darkness, and has translated us into the kingdom of his dear Son:
American King James Version×).
The apostle Peter, also acknowledging the centrality of Christ's role in the Kingdom, refers to it as "the everlasting kingdom of our Lord and Savior Jesus Christ" (2 Peter 1:11 2 Peter 1:11For so an entrance shall be ministered to you abundantly into the everlasting kingdom of our Lord and Savior Jesus Christ.
American King James Version×). Jesus Christ is our Lord and Master now, and He will reign supreme in the coming Kingdom (Revelation 17:14 Revelation 17:14These shall make war with the Lamb, and the Lamb shall overcome them: for he is Lord of lords, and King of kings: and they that are with him are called, and chosen, and faithful.
American King James Version×; Revelation 19:16 Revelation 19:16And he has on his clothing and on his thigh a name written, KING OF KINGS, AND LORD OF LORDS.
American King James Version×). As Savior of mankind, He is "the door" and "the way" by which we have access to God the Father and salvation in God's Kingdom (John 10:9 John 10:9I am the door: by me if any man enter in, he shall be saved, and shall go in and out, and find pasture.
American King James Version×; John 14:6 John 14:6Jesus said to him, I am the way, the truth, and the life: no man comes to the Father, but by me.
American King James Version×).