Matthew & Mark record Jesus’ message at the beginning of his ministry in similar ways. Mark records Jesus saying, “The kingdom of God comes near. Repent, and believe the good news!” While Matthew has “Repent, for the kingdom of heaven has come near.” (Both are from NIV.)
In a class I took in college, the Lutheran minister who was teaching the class said that Matthew probably didn’t use “kingdom of God” because he had a Jewish audience in mind who might take offense at “kingdom of God” so he changed it to “kingdom of heaven”.
Maybe Matthew’s original readers and hearers understood what he meant but sometime over 2000 years many people have understood “kingdom of heaven” as more of a prediction that the world is ending and judgement day is coming. (Repent! The end is near!) One biblical scholar shared in the forward of a book of his that I read that he understood Jesus was an apocalyptic preacher (the world is ending); and that since the world is still here almost 2000 years later, that that proved to him that Jesus was wrong, and so he “lost his faith”.
Another way of thinking of the kingdom of God or heaven coming is that we will relate to God in a different way. And that did happen due to Jesus ministry, death, and resurrection; establishing the church. There is also the destruction of the temple in Jerusalem and the Jews losing their homeland about 40 years later that would have been seen by them as the end of the world.