As with most film adaptations of literature, the movie version of The Crucible has many differences from the play, so I will focus on what I think are the most significant.
First, though the relationship between John Proctor and Abigail Williams is significant in the play, it is even more played up in the film. The movie depicts Abigail and John having a clandestine conversation in the woods, making it seem a bit more illicit than the parallel conversation in the play that takes place at Reverend Parris's house. Additionally, the movie depicts Abigail visiting John Proctor in his jail cell before she leaves town, telling him that she never meant for things to be the way they were, and that she only started the events in Salem because she wanted him. This never happens in the actual play. Instead, audiences find out through Reverend Parris that she has left Salem.
Certain events are also depicted differently in the film, highlighting the various grudges in Salem more visually than is depicted in the play (one of the luxuries film affords). In the play, the author, Arthur Miller, tells audiences directly of various land disputes, especially between Thomas Putnam and other people in Salem. The film allows us to visualize how each grudge led to an accusation by presenting it as a sort of montage.
Another way that the film shows viewers how the events are unfolding is when the camera cuts to the courts and shows the trials of Salem residents like Sarah Good and George Jacobs. The play, on the other hand, provides a second-hand account of these events through the conversation between Mary Warren and the Proctors.