The dynamics of each narrative are largely identical.
In the novel, a man journeys up a river - through violence and corruption - to find a man named Kurtz. The same is true of the film.
The character of Kurtz is essentially the same in both works. He is a genius who experiences a moral and psychological collapse, falling into a mental and spiritual chaos of his own creation after having failed to achieve a state of God-like individuation.
The protagonist experiences similar challenges to his own sanity in both stories.
The insanities presented in the book as stemming from isolation in the African jungle are in the film transposed to the jungles of Vietnam.
However, in the novel Marlow begins and ends as a relatively normal and sane person. In the film, the protagonist's relationship is established initially as a kind of kinship, defined by the protagonist's own psychological troubles. (Captain Willard is introduced in the film during a drunken rampage in his hotel room in Saigon.)
Additional differences can be found in the novel's interest in empire, capitalism, and commercial exploitation. These interests are significant in the novel while the film maintains a focus on psychological elements drawn from the book.
The question of what the alienation and loneliness of extended periods of time in a remote and hostile environment can do to men's minds is a central theme of the book.
This question is vital to the film and receives emphasis from the character of the photojournalist character. This character manages to discourse on Kurtz greatness while expressing the man's deeply developed "moral insanity".
Like Kurtz, the photojournalist becomes detached from conventional morality. Each work, the film and the novel, can be read as suggesting that conventional morality is the only kind of morality. Some moral agreement must exist to bond an individual to society. Without a social contract, people do not "go beyond" society to establish themselves as great individuals, as Kurtz seems to desire.
Rather, only moral chaos is encountered when all modes of moral thinking have fallen away. This is the "horror" that Kurtz discovers at the end of his life.
More topically, the novel is set in Africa and the film is set in Vietnam. The film features some light-hearted moments and some moments of tragedy that are only tangential to the central story line. These elements do not have much bearing in the novel, which is focused on the central story-line fairly thoroughly.
Race and "racial thinking" figure into both works but in different ways.
One more similarity: an air of doom is rendered in the style of both works.