They cannot be the same.

The Kingdom of God is spiritual , or "filled with spirits" or one/ a person is "full of the Spirit". A kingdom has a throne(Revelation 4). John describes what he sees..a throne with one sitting upon it. This is "Heaven", but not"The Kingdom of Heaven ". The synonamous usage of the two in the Gospels is in " Old Testament" times..no new Testament UNTIL the death of the Testator. No Holy spirit was given yet. No Kingdom of God within the heart yet.

 Kingdom of God

Acts 1: 3

³ To whom also he shewed himself alive after his passion by many infallible proofs, being seen of them forty days, and speaking of the things pertaining to the kingdom of God:

1 cor.15 :50

⁵⁰ Now this I say, brethren, that flesh and blood cannot inherit the kingdom of God; neither doth corruption inherit incorruption.

The Kingdom of heaven is physical, earthly, the Scepter of Judah, King Jesus ruling in righteousness and judgement.

Acts 1: 6

  ⁶ When they therefore were come together, they asked of him, saying, Lord, wilt thou at this time restore again the kingdom to Israel?

This is the Kingdom of heaven they are inquiring about.

Heb.1:8

⁸ But unto the Son [he saith], Thy throne, O God, [is] for ever and ever: a sceptre of righteousness [is] the sceptre of thy kingdom.

Psalm 145: 10-11

¹⁰ All thy works shall praise thee, O LORD; and thy saints shall bless thee.

¹¹ They shall speak of the glory of thy kingdom, and talk of thy power;

Just as Paul states that certain sinners will not inherit the Kingdom of God...he separates the two Kingdoms HERE:

Eph. 5: 5

⁵ For this ye know, that no whoremonger, nor unclean person, nor covetous man, who is an idolater, hath any inheritance in the kingdom of Christ( on earth) and of God( in the third heaven).

Of course flesh and blood cannot enter the third heaven. This statement of Paul here is three-fold. Kingdom of God only before the rapture, then Kingdom of Heaven at hand during the Trib. till the millennium..Kingdom of God and Heaven during the Millennium, and Kingdom of Christ and God in the fulness of times after the Millennium.

Kingdom of Heaven and God on earth

Rev. 21:1-3, 27

The New Jerusalem
Revelation 21:2

¹ And I saw a new heaven and a new earth: for the first heaven and the first earth were passed away; and there was no more sea.

² And I John saw the holy city, new Jerusalem, coming down from God out of heaven, prepared as a bride adorned for her husband.

³ And I heard a great voice out of heaven saying, Behold, the tabernacle of God [is] with men, and he will dwell with them, and they shall be his people, and God himself shall be with them, [and be] their God.

27²⁷ And there shall in no wise enter into it any thing that defileth, neither [whatsoever] worketh abomination, or [maketh] a lie: but they which are written in the Lamb's book of life.

Rev.22: 3-5

³ And there shall be no more curse: but the throne of God and of the Lamb shall be in it; and his servants shall serve him:

⁴ And they shall see his face; and his name [shall be] in their foreheads.

⁵ And there shall be no night there; and they need no candle, neither light of the sun; for the Lord God giveth them light: and they shall reign for ever and ever.