"The kingdom of the heavens"—the true rendering—is only named in Matthew. It is a dispensational term, while "the kingdom of God" is a moral thing. You find the terms used are in keeping with each individual Gospel. Matthew groups his subjects together dispensationally; Luke does so morally; both depart from the historic order, to which Mark keeps more than any of the others.
With a Jew the thought of the "kingdom of the heavens" was familiar. (See Deut. 11:21; Psalm 89:29; Dan. 2:44; 4:26-35, and other scriptures.) It is the rule of the heavens owned on earth. It was pronounced as "at hand," not as come, by John the Baptist (Matt. 3); by the Lord (Matt. 4); by the twelve (Matt. 10). The King was rejected; and in chapter 12, which ends the gospel to the Jew, the curse of antichrist is pronounced upon the nation, and a remnant owned who obey His Father's will. Then in chapter 13 the Lord begins a new action, as a sower; and the kingdom of the heavens takes a new character which the prophets did not contemplate: a sphere overrun with evil, and a mingled crop-the mysteries of the kingdom of the heavens. Instead of the true subjects taking their origin from Abraham, they do so from the word of God, which Christ sows, others accepting the authority of Christ nominally, as professors.
In Luke (Luke is the great moralizer), the term used is "kingdom of God," of which the Lord Jesus could say in answer to the inquiry of the Pharisees (they asked if it came with observation), that it was "in the midst of you" (Luke 17:21; J.N.D. Trans.), for God was there in Christ. Of the "kingdom of the heavens" it could only be said, it is "at hand"; and it did not (and could not) commence until the ascension of Christ. To have come in during His presence it would have been the kingdom of the earth, so to speak. His authority and that of the heavens was owned, even before the coming of the Holy Ghost, during the ten days of interval, by the disciples, who waited by His directions for that coming. It will run on in its present confused state until the Millennium; hence a good margin of time after the Church's history is over, as it had commenced before it.
There are two places where it gets a moral character from Paul-"The kingdom of God is not meat and drink; but righteousness, and peace, and joy in the Holy Ghost" (Rom. 14:17); "The kingdom of God is not in word, but in power" (1 Cor. 4:20). It is the exhibition or manifestation of the ruling power of God under any circumstances. A man must be born afresh to "see," or "enter in" to it, in the verity of it (John 3); not so of the kingdom of heaven, in which tares and wheat mingle. Souls may profess and submit to God's kingdom, as merely profession. Hence Luke 13:18 uses the term "kingdom of God" where nominal profession is noted in the parable, and where the "kingdom of the heavens" might be used interchangeably. Still, none but the saints would be really of it, as born again.
When the Millennium comes in, the present confused state of the kingdom of the heavens will be set aside by the judgment of the quick (living); and it will then be displayed in its verity in a twofold- heavenly and earthly-state of things. The Son of man gathers out of His kingdom; that is, the earthly part of it (see Psalm 8; Heb. 2), all stumbling blocks, and them that do iniquity. And then shall the righteous shine forth as the sun in the kingdom of their Father; that is, the heavenly sphere of it. (See Matt. 13:41-43.)