Although these terms are sometimes used interchangeably, strictly speaking they are not identical.
The Millennium is a period of time—1,000 years. The Kingdom of God is the reign of God and the family of God, which so far has two members—God the Father and God the Son, Jesus Christ. The Kingdom of God will rule the earth during and after the Millennium.
And the Kingdom will rapidly expand at that time. God's incredible plan is to increase His family by transforming human beings into literal children of God!
"But as many as received Him [Jesus Christ], to them He gave the right to become children of God, to those who believe in His name" (John 1:12 John 1:12But as many as received him, to them gave he power to become the sons of God, even to them that believe on his name:
American King James Version×).
God calls us His children from the moment that we receive the gift of His Holy Spirit (Romans 8:9 Romans 8:9But you are not in the flesh, but in the Spirit, if so be that the Spirit of God dwell in you. Now if any man have not the Spirit of Christ, he is none of his.
American King James Version×; Romans 8:14 Romans 8:14For as many as are led by the Spirit of God, they are the sons of God.
American King James Version×; Romans 8:16 Romans 8:16The Spirit itself bears witness with our spirit, that we are the children of God:
American King James Version×; Romans 8:21 Romans 8:21Because the creature itself also shall be delivered from the bondage of corruption into the glorious liberty of the children of God.
American King James Version×; 1 John 3:1-2 1 John 3:1-2  Behold, what manner of love the Father has bestowed on us, that we should be called the sons of God: therefore the world knows us not, because it knew him not.  Beloved, now are we the sons of God, and it does not yet appear what we shall be: but we know that, when he shall appear, we shall be like him; for we shall see him as he is.
American King James Version×). However, we become fully and permanently the transformed and glorified children God at the resurrection when Christ returns, when we receive a "spiritual body" (1 Corinthians 15:42-44 1 Corinthians 15:42-44  So also is the resurrection of the dead. It is sown in corruption; it is raised in incorruption:  It is sown in dishonor; it is raised in glory: it is sown in weakness; it is raised in power:  It is sown a natural body; it is raised a spiritual body. There is a natural body, and there is a spiritual body.
American King James Version×). That's when we "inherit the kingdom of God" (1 Corinthians 15:50-54 1 Corinthians 15:50-54  Now this I say, brothers, that flesh and blood cannot inherit the kingdom of God; neither does corruption inherit incorruption.  Behold, I show you a mystery; We shall not all sleep, but we shall all be changed,  In a moment, in the twinkling of an eye, at the last trump: for the trumpet shall sound, and the dead shall be raised incorruptible, and we shall be changed.  For this corruptible must put on incorruption, and this mortal must put on immortality.  So when this corruptible shall have put on incorruption, and this mortal shall have put on immortality, then shall be brought to pass the saying that is written, Death is swallowed up in victory.
American King James Version×).
During Jesus' earthly ministry, He preached "the gospel of the kingdom of God" (Mark 1:14 Mark 1:14Now after that John was put in prison, Jesus came into Galilee, preaching the gospel of the kingdom of God,
American King James Version×). Gospel means "good news." Jesus was preaching the good news that all human beings can become members of the Kingdom—the family—of God as well as what the rule of God's Kingdom will mean for the world's future.
When Christ returns, an angel will announce, "The kingdom of the world has become the kingdom of our Lord and of his Christ, and he will reign for ever and ever" (Revelation 11:15 Revelation 11:15And the seventh angel sounded; and there were great voices in heaven, saying, The kingdoms of this world are become the kingdoms of our Lord, and of his Christ; and he shall reign for ever and ever.
American King James Version×, NIV).