"Kingdom of heaven" is a term used exclusively in the Gospel of Matthew. The apostle Matthew used "kingdom of heaven" 32 times in his account about Jesus Christ's life and teachings. However, Matthew uses the terms "kingdom of heaven" and "kingdom of God" interchangeably. In Matthew 19:23-24 he uses the terms together in consecutive verses, clearly implying that they were synonymous. Often he calls it simply "the kingdom" (Matthew 6:13; Matthew 8:12; Matthew 13:19, Matthew 13:38; Matthew 24:14; etc.)
Comparing passages like Matthew 5:3 and Luke 6:20 shows that the terms meant the same thing:
"Blessed are the poor in spirit, for theirs is the kingdom of heaven" (Matthew 5:3).
"Blessed are you poor, for yours is the kingdom of God" (Luke 6:20).
Why did Matthew use the term "kingdom of heaven"? The Wycliffe Bible Dictionary explains that Matthew "was a Jew writing to his own race and respected their custom of using the name of God as little as possible and therefore spoke of the kingdom of heaven" ("Kingdom of God, Kingdom of Heaven," 1998).
The New Bible Dictionary agrees. "The use of 'kingdom of heaven' in Matthew is certainly due to the tendency in Judaism to avoid the direct use of the name of God. In any case no distinction in sense is to be assumed between the two expressions" (Second Edition, "Kingdom of God, Kingdom of Heaven," 1982).