Revealing mistake: In what is supposed to be an emotionally significant scene, Apollo offers his USA style boxing trunks to Rocky for him to wear in the upcoming rematch with Clubber. Problem is, there is one flashback scene when Rocky is training with Apollo that shows Clubber hitting Rocky with a body shot. You can see the top two or three inches of Rocky's trunks below the waistband and it is clearly the red, white, and blue trunks. In fact, the scene is from the second fight. You can see it in normal speed later.
Revealing mistake: When Apollo and Rocky are analyzing the "game film" of Rocky's first fight with Clubber, the film they are watching is actually movie clips of "Rocky III" itself. This is evidenced by the variety of angles and in-the-ring shots. In an effort to disguise this trickery, the "game film" was made black and white.
Revealing mistake: As the montage showing Clubber's rise to first-rank status is shown, various newspapers are shown detailing certain fights. When the shot of the paper with the article titled "Clubber Lang beats Jaffe in brutal 6th KO", look at the actual writing on the newspaper: It is nothing but about two paragraphs of meaningless, court-room style legal gibberish, and has absolutely nothing to do with boxing in any fashion. These two paragraphs are randomly printed over and over for every article shown on the paper.
Revealing mistake: During the rematch at the end of the film, Rocky is taunting Clubber with insults, and he reaches up and rubs Clubber's mowhawk a couple times to irritate him. The second time Rocky reaches for the top of Clubber's head, he can't quite reach it and Clubber just stands there and waits for an unrealistically long time for Rocky to reach the top of his head with his hand, and he makes absolutely no attempt to block or counter what surely would have been interpreted as a possible punch by any real boxer. This is certainly not the way any 'World Champion' would react in a real fight.