My understanding of Greek mythology indicated that Tartarus is a Greek version of hell, a place of punishment for wicked deeds. It also serve as imprisonment for the Titans, the early gods of the Greek pantheon (before that of the Olympians.)
Hades on the other hand, is not hell, but a place where all departed souls would reside. Hades is divided into several regions. Tartarus is only one region of Hades, and the deepest. Erebus is another. Most souls who are judged go to the Plain of Asphodel, another part of Hades, where there is neither sorrow or joy, and they don't retain their memory of when they were living.
Only the special ones go to Elysian Fields, a special part of Hades. Here the shades retained their memory of former lives. There is pleasure and joy here. The Elysian Fields is the closest thing to heaven, but it is not heaven because the gods don't dwell here (well, the Netherworld deites may live here, but it is still not heaven.)
Heaven to the Greeks is Olympus, and this is the home of gods, not a netherworld, thus not an abode of the dead. They associate Olympus with the sky or at least on the peak of the mountain.