Some say the "Kingdom of Heaven" refers to the a physical/political kingdom on earth while the "Kingdom of God" is the spiritual, coming reign of Christ.
Arguments against the two being the same often come down to hair splitting and misinterpretation of verses. For example, the site listed above relies on a single verse in an attempt to say they are different. In Luke 17:21, Jesus says "the kingdom of God is within you" (KJV). They then argue that since Matthew 11:12 says the Kingdom of Heaven suffers violence and is taken by force that the two cannot be the same (who can take the kingdom from within you?). The site only addresses where the terms are used interchangeably to blow them off by saying "they will be the same in the future."
However, it can be seen that Kingdom of Heaven and the Kingdom of God refer to the same thing. "Kingdom of Heaven" is found 31 times, only in the Gospel of Matthew. "Kingdom of God" is found 62 times in ten books of the New Testament (Matthew, Mark, Luke, John, Acts, Romans, 1 Corinthians, Galatians, Colossians, and 2 Thessalonians).
When the parallel accounts in the synoptic gospels are compared, whenever Matthew uses Kingdom of Heaven, the other will use Kingdom of God. Given the Jewish nature of Matthew's Gospel (Matthew never explains Jewish customs and traces the genealogy of Christ to Abraham where Luke, written for a gentile audience, continues back to Adam) and the Jewish habit of circumlocution in references to God*, "Kingdom of Heaven" would be preferable for the author and his intended audience.
*See especially the note about ultra-orthodox Jews never pronouncing the names.
Compare the following verses (all taken from the NET Bible):
Matthew 11:11 “I tell you the truth, among those born of women, no one has arisen greater than John the Baptist. Yet the one who is least in the kingdom of heaven is greater than he is. 11:12 From the days of John the Baptist until now the kingdom of heaven has suffered violence, and forceful people lay hold of it.
Luke 7:28 I tell you, among those born of women no one is greater than John. Yet the one who is least in the kingdom of God is greater than he is.”
Matthew 13:11 He replied, “You have been given the opportunity to know the secrets of the kingdom of heaven, but they have not.
Mark 4:11 He said to them, “The secret of the kingdom of God has been given to you. But to those outside, everything is in parables,
Luke 8:10 He said, “You have been given the opportunity to know the secrets of the kingdom of God, but for others they are in parables, so that although they see they may not see, and although they hear they may not understand.
Matthew 13:24 He presented them with another parable: “The kingdom of heaven is like a person who sowed good seed in his field.
Mark 4:26; He also said, “The kingdom of God is like someone who spreads seed on the ground.
Matthew 13:31 He gave them another parable: “The kingdom of heaven is like a mustard seed that a man took and sowed in his field.
Mark 4:30, 31 He also asked, “To what can we compare the kingdom of God, or what parable can we use to present it? It is like a mustard seed that when sown in the ground,
Luke 13:18, 19; Thus Jesus asked, “What is the kingdom of God like? To what should I compare it? It is like a mustard seed that a man took and sowed in his garden.
Matthew 13:33 He told them another parable: “The kingdom of heaven is like yeast that a woman took and mixed with three measures of flour until all the dough had risen.”
Luke 13:20, 21; Again he said, “To what should I compare the kingdom of God? It is like yeast that a woman took and mixed with three measures of flour until all the dough had risen.”
Matthew 18:3 and said, “I tell you the truth, 1 unless you turn around and become like little children, you will never enter the kingdom of heaven!
Mark 10:15 I tell you the truth, whoever does not receive the kingdom of God like a child will never enter it.”
Luke 18:17; I tell you the truth, whoever does not receive the kingdom of God like a child will never enter it.”
In each instance, Matthew used the phrase “kingdom of heaven” while Mark and/or Luke used “kingdom of God.” The obvious conclusion is that the two phrases refer to the same thing.
Within Matthew, the terms are used interchangeably on occasion. After the rich young ruler has left, Jesus says to the disciples, 19:23 Then Jesus said to his disciples, “I tell you the truth, it will be hard for a rich person to enter the kingdom of heaven! 19:24 Again I say, it is easier for a camel to go through the eye of a needle than for a rich person to enter into the kingdom of God.” Clearly, even Matthew considers the terms parallel.