The main difference is in the physical location and its meaning for the characters. Heart of Darkness is set in the Congo, during European Colonization, and concerns the monetary desires of the trading companies and the affect that the primitive, harsh jungle has on the people who enter it. Apocalypse Now is set during the Vietnam War, and although the Vietnam Jungle has a similar effect on the people, it is for different reasons; the Vietnam War destroys men through its brutality and the things that men will do when given governmental carte blanche, not strictly through their own moral deterioration as in the novel.
"Going up that river was like traveling back to the earliest beginnings of the world, when vegetation rioted on the earth and the big trees were kings. An empty stream, a great silence, an impenetrable forest. The air was warm, thick, heavy, sluggish. There was no joy in the brilliance of sunshine."
(Conrad, Heart of Darkness, gutenberg.org)
The settings are perhaps more similar than different; they are both jungles where seemingly civilized men become brutal and evil, perhaps feeling that the locality removes the need to be moral, and that their actions in this far-off land will have no consequences in the "real world." However, it is the justification for action that differs: Kurtz in Heart of Darkness has been affected by some unknown part of the jungle--something that Marlow feels he has just glimpsed the edge of--and has lost himself in power and personal excess; in the film, Apocalypse, Kurtz has been broken by the horrors of the Vietnam War, and feels that his actions are as morally justified as any, since the seemingly-moral government that sent him into the jungle has justified the war's atrocities.