Right before the main event, the channel that is presenting the fight between Balboa and Dixon throws up some stats for the two fighters on the screen. The headshot they use for Rocky is taken from the upcoming fight that Rocky has not yet fought in.
When Rocky gives little Marie a ride home, it shows him getting in the van and the van automatically starts, his hand never goes to the ignition to put the key in. Then when he gets out to go talk to the drunk, it shows him turning van off.
At the end of round two, when the bell rings, Rocky's trainer Duke jumps through the ropes and falls into the ring. The towel on his shoulder hangs on the ropes. The next scene showing Rocky in his corner has Duke going through the ropes again with the towel still on his shoulder.
When Rocky and Marie was walking toward the van from the bar, the van is parked about a quarter of a block away from the bar. After Rocky come back from confronting with the drunk, the van is parked on the corner and blocking the crosswalk.
When Rocky is talking to Paulie in the restaurant about the simulated fight, the bottle on the bar changes from one with a black label on the neck, to one with a blue label on the neck, back to black, then back to blue.
When Rocky applied for a boxing license in Rocky V, he was denied, as the doctors stated he had irreversible brain damage. In this movie, according to the licensing board, Rocky passed all of his medical exams with "flying colors." They cannot both be correct at the same time. However, Sylvester Stallone has addressed this in interviews, stating that the standards of acceptable health for boxing could have changed over time - and Rocky IS completely fit by modern standards. Additionally, he never got a second opinion and yielded to Adrian's advice to quit to protect his health.
During the scene when Steps is looking over Rocky's collection of championships and photos, it has title belts on the wall that Rocky never actually won in his career. However, 16 years have passed between Rocky V and this film (in which we don't see the events), and it is entirely possible that, in the time between films, Rocky managed additional fighters and THEY won the awards in question.
After the pre-fight prayer with Spider Rico, Rocky calls Spider Rico 'Pedro'. Spider Rico is played by Pedro Lovell. "Spider" is clearly a nickname, so it is possible the character's name is Pedro "Spider" Rico. As Tony Burton plays a character named Tony "Duke" Evers, and many other actors in the Rocky movies use their given names as their character names, this is not outside the realm of possibility.
When Rocky Fights Dixon it shows his record as 57-23-1. He was 44-20 going into his first match with Apollo. He lost to Apollo, then beat him, then had 10 title defense victories. His record at that point was 55-21. After splitting with Clubber Lang he was 56-22. One more fight against Drago makes 57-22. The draw could have been the fight with Thunderlips in Rocky III, but where did the additional defeat come from?