I have to write an essay comparing and contrasting an Oedipus the King film with the play. I watched the 1986 BBC version of Oedipus the King, but I am a little stumped trying to come up with comparisons and differences.

Instead of the BBC version, I would suggest you take a look at two films from the 1960s: Oedipus Rex (in Italian, Edipo Re), directed by Pier Paolo Pasolini, and Oedipus the King, starring Christopher Plummer, Orson Welles, and Lilli Palmer, and directed by Philip Saville. Both of these, especially Pasolini's version, are more cinematic than the 1986 one to which you refer, which is basically a filmed stage-play with the one novelty of being performed in modern dress.
In Pasolini's film, not surprisingly for him, there are several unusual, offbeat features. The story is told beginning not as Sophocles does—with Thebes in crisis and Oedipus already king—but with Oedipus's earlier history and childhood, showing his infancy, his rescue by the shepherd, and his youth in Corinth. The emphasis is more visual than verbal. Sophocles tells the details not only in flashback, but in the words of the characters themselves—the Messenger, the Shepherd, and of course Oedipus and Iocasta—as they recall the details and reconstruct the tragic story. A filmmaker, by not actually showing these happenings on screen, would not be using the full resources of his medium. Therefore, a genuinely cinematic treatment would have to diverge from Sophocles to the extent that it would depict things visually that are not shown on stage in the play but are merely narrated.
Similarly, the incident at the "triple road," in which Oedipus unknowingly kills his father, is one of the most dramatic points of the story. In the version starring Plummer, we actually see this happen, rather than simply being told about it.
Pay close attention to the music in Pasolini's film, too. As he typically does, he provides an eclectic mix of background scoring, ranging from Mozart to Japanese-style music that would sound appropriate in a Kurosawa film. In addition, he updates the setting.
You might ask yourself what these kinds of changes do to the effect the play has on the audience, even if they do not actually alter the basic storyline of the tragedy.

By and large, this version is quite faithful to Sophocles’ drama. The filmmakers make no major revisions to the story nor do they alter themes, so I can see why this part of the question is hard to answer. There might be minor differences between the film’s translation of the original text and the translation of the print version you are using, so some little nuances might exist on a line by line basis.
The biggest change this TV version makes is the way the setting is portrayed. Most stage versions keep the action set in the ancient world. Some even retain the ancient Greek theater tradition of having the actors wear masks.
Instead of giving the audience an ancient Greek setting, the filmmakers mix and match elements from different time periods in the costuming and sets. While some characters wear robes right out of antiquity, other characters dress like medieval peasants or in twentieth century dress. For example, Oedipus wears a white suit, resembling a nineteenth century European monarch more than an ancient king.
This anachronistic aesthetic blend is likely meant to make the story seem more timeless, showing how the themes of guilt, truth, and pride are relevant no matter the time or place.