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).
God calls us His children from the moment that we receive the gift of His Holy Spirit (Romans 8:9; Romans 8:14; Romans 8:16; Romans 8:21; 1 John 3:1-2). 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). That's when we "inherit the kingdom of God" (1 Corinthians 15:50-54).
During Jesus' earthly ministry, He preached "the gospel of the kingdom of God" (Mark 1:14). 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, NIV).