The idea of a kingdom in the biblical sense is not just a sphere under the royal power of a monarch, but a living supra terrestrial life of which God sits on a throne as a redeemer, priest, prophet, judge and king. All in this sphere are redeemed by his blood and born into this invisible realm by faith. Jehovah in Christ is their God, master and royal Lord. Christ is that single person who possesses everything under his rule and so he is filled with royal glory.
The idea of God's kingdom is related to those images found in David and promised in Messiah, but it is God himself in his divinity that is the root of the definition:
And they were calling to one another: “Holy, holy, holy is the Lord Almighty; the whole earth is full of his glory.” At the sound of their voices the doorposts and thresholds shook and the temple was filled with smoke. “Woe to me!” I cried. “I am ruined! For I am a man of unclean lips, and I live among a people of unclean lips, and my eyes have seen the King, the Lord Almighty.” (Isaiah 6:3-5, NIV)
As far as the term 'kingdom of God' verses 'kingdom of heaven' there is no difference. Mathew wrote primarily to a Jewish audience that preferred to avoid the term God, as it was taboo to speak his holy name. However, the other evangelists seem to prefer 'kingdom of God' as Gentiles would more easily understand the Messianic aspect from that term, rather then revert to their own philosophical notions of 'heaven'.