This is probably the source of most people's knowledge of Becket.  It was the first place I heard of him, when we watched the movie in my high school history class.  I was impressed by the drama of the story, but at the time I didn't think much about it.  Years later, though, my interest revived, until I ended up doing this.

The story in the movie is the same as the story in the play, the synopsis of which can be read here.  There are some modifications though, as follows.

In the movie the shade of Thomas doesn't appear.  Henry just talks to himself, remembering.

Gwendoline doesn't sing the ballad of Thomas's parents, so there's no discussion about that.  She sings something in Welsh instead.

In the play, Thomas takes the Saxon girl's brother outside with him and takes his knife away.  In the movie the brother is hiding in the woods and attacks Thomas.

The monk, who's just called "the little monk" in the play, gets a name: John.  And he also attacks Thomas at their first meeting.  What is it with all these knife attacks?  Is it really clumsy foreshadowing?  Here the monk hates Thomas at first but starts liking him after he realizes he's sincere in his archiepiscopal devotion.

Certain scenes from the play are omitted in the movie: the bishops' discussion of Thomas after he's left to hunt with the king, the wait in the sacristy in France (except for the message about the old archbishop), the trip across the Channel back to England.  Certain scenes are added: the banquet, Thomas's consecration (with Gilbert Foliot officiating, of all people.  What happened to Henry of Winchester?), Thomas's declaration that he'll appeal to the pope when he's been accused of embezzlement (we're just told about it in the play), the escape to France.  Also, in the movie Thomas actually speaks to the pope, while in the play the pope and cardinals just talk about him.

In the play the stage directions have the knights attacking Thomas, the lights going down, and then the lights coming back up again onto the monks beating Henry with the same motion the knights had been using.  They didn't do this in the movie, and I don't know why.  It seems like it would be a good and effective transition.

Some of the clothing is right for the time period.  Henry's tunic when he rides into his captured town is an excellent example, as is his tunic at the final dinner.  But a lot of the clothes come straight from the fifteenth century.  This includes the servants at the banquet, but the queens are the most egregious offenders.  Their headdresses are obviously not twelfth century, and Eleanor's robes as well.  I don't know if the costumers were going for a general medieval look, or thought these clothes were more impressive (fifteenth-century headdresses are more impressive than twelfth-century veils), or what.  Thomas is even wearing a huque, of all things, in France.  The guards' armor is all right, but when the princes fight each other they're wearing close helmets.  Even a couple of things in the archbishop's vestments aren't period.  Thomas makes a point of telling the monk to do up all the little ties on the side, but at that time the sides were solid.  And there's his miter.  The points of the miter used to be on the sides, not front and back, but the style changed soon after Thomas's death, so even medieval depictions show him with the front and back points.  The ecclesiastical things bother me least of all, though.  There's a point where being overly accurate on the details is probably just going to confuse the viewers.  In fact, most people probably don't care about any of the costuming.  I'm just nitpicky.

When Thomas and Henry go hunting and get caught in the rain, the sun is obviously still shining.  I have seen that kind of weather really happen, but it's still kind of funny.

The music is a little heavy-handed at times, blaring at dramatic moments.

Accused of wrongdoing, Thomas says, "As head of the Church of England, I forbid you to pass judgment on me," and appeals to the pope.  What a curious way to phrase it.  It was my understanding that the Church of England (as opposed to the Church in England) wasn't around till Henry VIII.

Someone needs to teach Brother John to be a proper crucifer.  He carries the metropolitan cross as casually as if it's a stick or something.  It looks undignified.  Going down steps is no excuse for holding it crooked.

At the end no one is visible in the cathedral except Thomas, John, and the knights, although the monks singing vespers are audible.  I suppose it focuses the attention on the main characters, but it makes it seem like the action's happening in a void.

The knights use their swords to stab, not hack, which seems an inefficient way to use broadswords.  And Thomas's holy tonsured head, that everyone made such a big deal about in the Middle Ages, is untouched.

I'll end with the thing that pleased me most: Thomas begins vespers by singing "Deus in adiutorium meum intende."  (God, come to my assistance.)  Leaving aside the fact that the historical Thomas wasn't officiating - it's the actual unfinished Vespers they were singing that night!
 

As many costume links as one could desire.