The Bible begins with God’s creation of the heavens and earth and ends with the creation of a new heaven and earth.
Then I saw a new heaven and a new earth; for the first heaven and the first earth passed away, and there is no longer any sea. 2 And I saw the holy city, new Jerusalem, coming down out of heaven from God, made ready as a bride adorned for her husband. 3 And I heard a loud voice from the throne, saying, “Behold, the tabernacle of God is among men, and He will dwell among them, and they shall be His people, and God Himself will be among them, 4 and He will wipe away every tear from their eyes; and there will no longer be any death; there will no longer be any mourning, or crying, or pain; the first things have passed away.” 5 And He who sits on the throne said, “Behold, I am making all things new.” And He said, “Write, for these words are faithful and true.” Revelation 21:1–5 (NASB95)
The latter version is far superior to the first, just the New Covenant is far better than the old. Adam and Eve followed the Deceiver into sin and corruption, thus defiling God’s creation. Foreseeing all this, God constructed the initial heavens and earth with a solid firmament, which separates God’s 
domain from ours. This amazing structure restricts access to the third heaven, the location of God’s throne. It is only via the New Covenant that we gain access to God’s heavenly throne and presence (Hebrews 4:16), where one day we will go bodily after the resurrection.
The new heavens and earth does not seem to have a firmament or separator. The Bible says it will not even have a sun because God’s light and glory, now concealed by the firmament, will provide all the illumination we need. The New Jerusalem will be part of this new creation, but it will not be part of the old order of things. It is not going to be a restoration of the physical Jewish kingdom. That has long since ceased to be significant. The New Jerusalem will be made up of born-again believers, the true Israel of God.
I saw no temple in it, for the Lord God the Almighty and the Lamb are its temple. 23 And the city has no need of the sun or of the moon to shine on it, for the glory of God has illumined it, and its lamp is the Lamb. Revelation 21:22–23 (NASB95)
The purpose of the new heaven and earth will be to provide a perfect habitation for God’s people, one in which God manifestly dwells with us continually.
God’s purpose from the beginning will be finally fulfilled. He will dwell among his people.
