There is no difference in meaning between the phrases "kingdom of heaven" and "kingdom of God".  The gospel of Matthew is the only book that uses the phrase “kingdom of heaven” and he uses the phrases interchangeably.  Luke and Mark only use the phrase “kingdom of God.”

The following scriptures show that the expressions are the same and interchangeable:

Matthew 4:17: "Jesus began to preach and to say, Repent, for the kingdom of heaven is at hand."  Mark 1:14, 15 reads, "Jesus came into Galilee preaching the gospel of the kingdom of God, and saying, the time is fulfilled, and the kingdom of God is at hand."

Both scriptures refer to same time period, when Jesus began preaching after John the Baptist had been arrested. The phrases are synonymous.

Matthew 13:11: "It is given unto you to know the mysteries of the kingdom of heaven."     

Mark 4:11: "Unto you it is given to know the mystery of the kingdom of God."   

These two scriptures tell of Jesus’s explanation of why He talked in parables.

These Matthew scriptures also use both phrases showing that they have the same meaning:

Matthew 19:23-24,  “And Jesus said to his disciples, ‘Truly, I say to you, it will be hard for a rich man to enter the kingdom of heaven.  Again I tell you, it is easier for a camel to go through the eye of a needle than for a rich man to enter the kingdom of God.’”   The thought is expressed as “kingdom of God” in Mark 10:24, 25, and Luke 18:24 and 25.

Here are a few more scriptures:

Matthew 13:31: "The kingdom of heaven is like to a grain of mustard seed." Mark 4:30, 31: "Whereunto shall we liken the kingdom of God…it is like a grain of mustard seed." Luke 13:18, 19: "Unto what is the kingdom of God like? It is like a grain of mustard seed."

These parallel scriptures show how the different gospels use the phrases “kingdom of heaven” and “kingdom of God” to refer to the same concept even when spoken of at different times during Jesus’ ministry.