The kingdom of God and the kingdom of heaven seem to be variations of the same idea. A kingdom implies a king. Our king is Jesus. Jesus said that His kingdom was not of this world (John 18:36). Jesus' authority did not come from man but from God (Luke 22:29).
Entrance into the kingdom of God is by a new birth (John 3:5), repentance (Matt. 3:2), and the divine call (1 Thess. 2:12). We are told to seek the kingdom of God first (Matt. 6:33) and to pray for its arrival (Matt. 6:10). "The kingdom of God is not eating and drinking, but righteousness and peace and joy in the Holy Spirit," (Rom. 14:17). It is also a future kingdom where full rulership in the actual presence of the King Jesus will occur when He returns to earth.