My purpose here was to stimulate conversation. It's been stimulated. I have settled the issue for myself and have a peace about it. If you still haven't answered the question for yourself, then you need to keep studying until you get either "they're the same" or "they're not the same".
I'll give you a hint. Stay in Matthew 13 until you understand all the parables and what they mean. For example:
What is the "treasure"?
What is the "Pearl of great price"?
How can tares be among the wheat in the kingdom?
In the end, the concept of the "kingdom of heaven" is more than the concept of the "kingdom of God" but includes that also. It's that "more" that makes them different. A couple of posters have touched on it in this thread.
The Kingdom of God is whereever God rules and reigns. Jesus said, "The Kingdom of God is within you." That is, God is ruling and reigning in the hearts of His people. He has gone in, made His abode in their hearts, and freed them from the bondage of sin.
People have always misunderstood the Kingdom of God. The Jews at the time of Christ did. Many Christians today do as well. The reason is, people think about His Kingdom like they think of the kingdoms of men. The two are different. Christ's Kingdom is a spiritual Kingdom, unlike anything that has ever existed. It is a present reality, with Him ruling it presently. There is coming a day when we will enjoy it to a fuller extent because Christ is going to wipe everything that offends off this earth, every other kingdom will be destroyed, and all that will be left is Christ's Kingdom on the New Earth for all eternity.