There are probably too many differences between the book and the film to list and explain in any kind of concise manner. If I had to pick a single main difference between the two formats for Chris McCandless's story, I would have to choose the tone that each format uses to explore McCandless. Krakauer's book is more of an examination of McCandless's life and death. Krakauer gives childhood background information about McCandless that the movie doesn't go into. That's no fault of director Sean Penn. Movies generally have to leave out a lot of detail in order to fit a rough two-hour time limit.
At other times, Krakauer tries to explain McCandless's actions and motivations by comparing him both to other men that died alone in the wilderness and to Krakauer himself. These comparisons are also something that the movie doesn't bother with.
The book is interesting because the reader often gets the impression that Krakauer is empathetic toward McCandless while also trying to stay objective. On the other hand, the movie's tone doesn't try to objectively analyze McCandless. The big difference in the movie is that Penn seems to want to celebrate McCandless and his individualism. It's possible that Penn steered the film in this direction because he wanted to show audiences McCandless's story from McCandless's perspective. Krakauer's account is going for a more neutral, omniscient narrative point of view. For example, Penn's film shows McCandless as being a bit more indifferent to government rules concerning where he can go in the wilderness and what he can do. Krakauer's account uses dialogue from interviews he conducted with people that knew McCandless, so we see McCandless as a rebellious and angry person rather than a romantic of sorts:
Hell, no. How I feed myself is none of the government's business. Fuck their stupid rules.