No wonder Widmark and Wayne didn't get along.
You'll usually read that it was about politics but my understanding through the gossip mill is that it was because Widmark angered Wayne by taking liberties with the script. Remember the scene in which Crockett and Bowie first meet? Bowie comes to Crockett's aid during a street brawl with Emil Sand's thugs, and soon the only two thugs still standing are the two Bowie has in dual headlocks. In the scene as we know it today, Crockett borrows Bowie's knife and uses the handle as a bludgeon to knock one of the thugs out. Bowie and Crockett exchange pleasantries and then Bowie takes his knife back and knocks the second guy out.
But Widmark wanted to show a more savage Bowie, and when the scene was originally shot, after Crockett borrows the Bowie knife and knocks one of the thugs out, Bowie says, "No, not like that . . ." and then, taking his knife back, he says, " . . . like THIS!" and slashes the second thugs throat. Then he says, "Wait a minute, Davy, while I finish this." And then, laughing maniacally like Widmark's Tommy Udo in KISS OF DEATH, Bowie runs around stabbing to death all the other unconscious thugs.
Wayne was incensed because he had to re-shoot the scene, and their argument about it caused a permanent rift. Swear to God.