Because we now know (*) that the Gospels of Matthew and Luke were substantially based on Mark's Gospel, and because the Gospels of Matthew and Luke have additional material in common, believed to have come from the hypothetical 'Q' document, we can compare the three gospels and look for passages where other gospels use 'kingdom of God' where Matthew uses 'kingdom of heaven'. If such passages exist, they will point to there being no material difference between the two terms, with the choice of 'kingdom of heaven' simply being a stylistic difference.
One parallel in which Mark was the original for the passage in Matthew is Matthew 13:11, which uses 'kingdom of heaven', while the original in Mark 4:11 uses 'kingdom of God':
Matthew 13:11: He answered and said unto them, Because it is given unto you to know the mysteries of the kingdom of heaven, but to them it is not given.
Mark 4:11: And he said unto them, Unto you it is given to know the mystery of the kingdom of God: but unto them that are without, all these things are done in parables:
One parallel in which Matthew and Luke have similar passages, from the 'Q' document (where minor differences in wording exist, most scholars believe Luke is closer to the original in Q):
Matthew 5:3: Blessed are the poor in spirit: for theirs is the kingdom of heaven.
Luke 6:20: And he lifted up his eyes on his disciples, and said, Blessed be ye poor: for yours is the kingdom of God.
Examples such as these show that the author of Matthew chose to use 'kingdom of heaven' even when copying a passage that used the term 'kingdom of God'.
However, sometimes our author chose to use 'kingdom of God', in line with his sources, for example:
Matthew 19:24: And again I say unto 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.
Mark 10:25: 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.
(*) John Dominic Crossan, in The Birth of Christianity, page 110-111, speaks of a massive consensus among scholars in favour of Markan priority. He says there is a smaller, but still substantial majorityof scholars who believe that the hypothetical 'Q' document was the source of other material on which Matthew and Luke agree.