Our Father which art in heaven,
Hallowed be thy name.
Thy kingdom come.
Thy will be done, in earth as it is in heaven ...
-- Matthew 6:9-10
There is no doubt that heaven is the realm in which the Father dwells, where its citizens prefer to do His will, and that earth is the realm in which the Father does not dwell, and where its citizens prefer to do their own will.
Sheol is the place of the dead, which is part of this realm. Clearly, the Father doesn't dwell in Sheol.
At the beginning of the biblical narrative, in Genesis, we are told that the "Tree of Life" was in the garden of Eden from which Adam and Eve were expelled, having chosen their own will over the Father's. This, of course, would have brought chaos to Eden for eternity, so, the Father considered it better to place them outside its gates in a realm of their own, which they could trash if such was their will.
Putting them out of Eden gave them something that they did not possess when they where within -- the capacity to discover for themselves which is best: the Father's will, or their own.
And the LORD God said, Behold, the man is become as one of us, to know good and evil: and now, lest he put forth his hand, and take also of the tree of life, and eat, and live for ever: Therefore the LORD God sent him forth from the garden of Eden, to till the ground from whence he was taken. So he drove out the man; and he placed at the east of the garden of Eden Cherubims, and a flaming sword which turned every way, to keep the way of the tree of life.
-- Genesis 3:22-24 (KJV)
At the conclusion of the biblical narrative (according to the Christian canon), in Revelation, readers are told that the "Tree of Life" will be in the paradise of God, from which all overcomers will be invited to eat.
He that hath an ear, let him hear what the Spirit saith unto the churches; To him that overcometh will I give to eat of the tree of life, which is in the midst of the paradise of God.
-- Revelation 2:7 (KJV)
Readers of Revelation are also told the purpose the Tree of Life will serve:
And he shewed me a pure river of water of life, clear as crystal, proceeding out of the throne of God and of the Lamb. In the midst of the street of it, and on either side of the river, was there the tree of life, which bare twelve manner of fruits, and yielded her fruit every month: and the leaves of the tree were for the healing of the nations.
And how one is to overcome and, thus, obtain access to the Tree of Life:
Blessed are they that do his commandments, that they may have right to the tree of life, and may enter in through the gates into the city.
-- Revelation 22:14 (KJV)
When the thief closed his eyes on this realm, he would open them again in the next (1 Corinthians 15:51-52). Any time that might pass for humanity between closing and opening his eyes would be unnoticed by him. When the thief does opens his eyes, however, he will do so in a different body (suitable for eternity) on a different earth, with a different heaven (canopy of stars), and Jesus will be there as his King and Lord.
Jesus, of course, had some loose ends to tie up on earth before he could join the thief in heaven, all of which would take place before the thief opened his eyes. Jesus is not constrained by time and space and is freely able to move between the heavenly and earthly realms.
Is there a difference between Heaven and Paradise?
No. According to the biblical narrative Heaven, Eden, Paradise, and the City are all terms that distinguish the realm of the Father from the realm of men, i.e. the realm of peace, prosperity and life from the realm of chaos, destruction and death.
The more hearts in the realm of men that are inclined towards the commandments of God, the more it will resemble the realm of the Father, which He has declared is His will. Such a state, however, is not what is prophesied for the realm of men.