Heaven is the term we generally use when we're thinking about a "place" where God is thought to live. It's also the place where we believe that we will ascend to upon our death. Many people envision this being a place above this created world with its opposite, hell, being located in the underneath.
The “kingdom of heaven” is a term used frequently by Jesus, alhough he never clearly defines it. He mostly refers to it in parables. It's also sometimes called the “Kingdom of God.” The concept is one of the central messages of Jesus’ preaching. It is a symbolic term calling up the Jewish belief that God created this world, and is its universal king and judge who constantly works for the good of the people. But unlike the concept of a traditional kingdom, this kingdom isn't a physical one but one that encompasses a reign of justice, peace, and love — in short, a paradise. Jesus also repeatedly pointed out that “The kingdom of God is among you” and encouraged his followers to work to spread the Kingdom of God. To follow that example of Jesus suggests that it's not something coming in the future, but something to be lived in the present. We can use our gifts to further a world of justice and peace right now, in this world.