This question is tricky because there are two film versions of The Ransom of Red Chief. Without knowing exactly which movie you are viewing, it is difficult to provide a completely accurate answer. I am going to assume that your assignment is on the color 1998 version rather than the black and white 1950s version. (It is a shame. The 1950 version is closer to the short story, but it is difficult to find.) The differences between the book and film of The Ransom of Red Chief are substantial.
Probably the first thing you notice upon watching the movie is that the narrator has shifted. The sheriff is the narrator in the movie; Sam is the narrator in the book. Sam and Bill enter the story on a train in the movie rather than in a buggy. The location of the movie changed. In the movie, they are in Missouri, while the book takes place in Alabama. In the movie, the fraudulent real estate deal seems to have Sam and Bill the victims, while in the book they are trying to get money to pull off a fraudulent real estate deal.
Second, the physical aspects of Johnny (called Andy in the movie) are not the same. In the book Johnny is a redhead, but in the movie he is blond. In addition, the film version of "Johnny" (Andy) is quite a bit older than the book version. The book states that he is six to seven years old. In the movie, he is ten. This change in age affects the reaction to Johnny's antics. A six or seven year old might be considered hyperactive and precocious, a little boy who does not know any better. By choosing an older child, Johnny is perceived more as an uncontrollable spoiled brat. Also, in the short story there is no mother mentioned, yet in the movie she is a major character. She participates equally with Mr. Dorset in most of the scenes.
The kidnapping is different, too. In the short story, Johnny is throwing rocks at a kitten when he is kidnapped. He ends up at the bottom of the buggy after fighting with Sam and Bill. In the movie, he is "running away from home" and is talked into voluntarily going with Bill and Sam as part of a deal. In the movie Bill and Sam hide out in the mine they purchased rather than a cave, like in the short story.
Another departure from the original story is Johnny/Andy's antics. The sequence where Sam and Andy deal with the ghosts in the cave is different from the short story, where Johnny mostly deals with physical pranks rather than psychological and verbal ones. The ransom is also different—in the movie, they ask for forty-seven dollars and fifty cents rather than two thousand dollars, as in the book.
Much of the ending is completely different, dealing with disguises, escaped prisoners, the horse chase, Mr. Dorset working with Sam, the bomb, and the Dorset's supporting Sam and Bill leading them to become heroes rather than paying Mr. Dorset at the end of the story. None of these sequences appear in or are even remotely connected to the short story. The ending of the movie lacks the switch where Bill and Sam pay Mr. Dorset to take Johnny back. Instead, Bill and Sam become part of the community.
There is a saying that a movie is never as good as the book. In this case, it is correct. The changes to the story, I would assume to make it more entertaining, take away from the beauty of O. Henry's story and do not add to the story at all.
I believe the 1950 movie version you refer to is the one contained on a DVD titled O. Henry's Full House. It stars Fred Allen and Oscar Levant. It was well done, and, as you say, more faithful to the original short story. O. Henry's Full House is readily obtainable through Netflix. The other four stories on the disc are also well done, especially "The Gift of the Magi" starring Charles Laughton.
"Top movie talents lend their gifts to this 1952 anthology of O. Henry short-story adaptations, introduced and narrated by Nobel laureate John Steinbeck. "The Gift of the Magi," "The Clarion Call," "The Cop and the Anthem," "The Last Leaf" and "The Ransom of Red Chief" come to life, thanks to an impressive roster including Anne Baxter, Richard Widmark, Marilyn Monroe, Charles Laughton and directors Howard Hawks and Henry Hathaway."