When Rocky turns the TV on for the turtles, Brent Mussberger is talking about the Rams and their field goal kicker, Frank Corral. The plot of the movie has the time line at mid to late 1976 (Apollo's trainer, Duke, mentioned a couple of scenes earlier that about 6 months had passed since the first fight), there is a problem because Frank Corral did not play for the Rams until 1978.
The film takes place from New Year's Day to Thanksgiving Day, 1976. However, when Rocky is at the old Pontiac dealer at Broad and Pine Street in Philadelphia, the green car in the showroom and his new car are 1979 models.
When Apollo is in the office attempting to set up a rematch, he calls Tony Burton's character by the actor's name. At this point he wasn't credited as a named character, but starting in Rocky III he's known as Duke, not Tony.
In the first fight depicted in the first film, which also shown in the beginning of this film, Rocky's right upper lid gets hematoma and they cut it. And when Mick tests Rocky's sight, we clearly see that it's the right eye that has vision problem. But during second fight, when Rocky is first knocked-down the commentator says that Apollo has hit the damaged left eye, which was cut in the previous match.
When Adrian is in the coma, Rocky's facial hair is used to give a general idea of time. Just before Adrian comes out of the coma you can Rocky with almost a full beard. The shot pans over to Adrian opening her eyes. When the shot pans back to Rocky, he barely has a 5 o'clock shadow.
When the priest is marrying Rocky and Adrian, he says Adrian's last name like "Pennino". But according to the name listings at the press conference in [Rocky Balboa] (based on what we see on Paulie's card), the name is actually Panina.
The final fight between Rocky and Apollo takes place in November. We are told that this fight is taking place exactly 10 months after the events of the first film, This places the fight from the original Rocky in the January of the same year. However when Rocky is about to begin his entrance into the arena he comments on his robe,comparing it to the one he wore for the first fight, saying it fits well "unlike last years", Despite the fact that both fights took place in the same year.
When Rocky and Apollo are on the canvas at the end of the fight, Adrian and Paulie are at home standing up in front of the TV yelling for Rocky to get up for the entire count. When he finally gets up, they show them jumping up out of their chairs as if they have been sitting through the count.
When Apollo is on his way to the ring, his robe is done up and a towel is wrapped tightly around his neck. The scene cuts away to a shot of the ring and then cuts back to Apollo and his robe is now undone and the towel is dangling from his neck with his chest visible. The scene cuts away again and then cuts back to Apollo and his robe is done back up and the towel wrapped tightly around his neck again. It appears that they used footage from 2 different takes, one with the robe done up and the other with the robe undone.
During the "Gonna Fly Now" training scene. Rocky is running through the city. He makes a left turn down a very crowded street. As he turns there is a golden coloured Gremlin parked on the opposite side of the road. In a subsequent front-on shot he is shown running much further down the road but he passes the same Gremlin once again.
As the ambulance is headed toward the hospital, it is being closely followed by a news van, no more than one second behind. When the ambulance makes a right turn down a side street, the news van is nowhere to be seen. But when they pull up to the hospital, the news van is right behind the ambulance again.
During the fifteenth round, Rocky lands a left hook and Apollo Creed is knocked down. Rocky falls as the result of his own punch. The referee begins to count both fighters out. The referee would only be counting Apollo Creed out because Rocky's punch knocked him to the canvas. According to the rules of boxing, Rocky should not be in danger of being counted out because he was not struck by a punch. Rocky was off balance and so the referee would rule it as a slip.
When Rocky & Adrian meet with the real estate agent to buy the Townhouse, as they're about to walk in the front door, Rocky taps on the address numbers and says that they almost add up to 9 which is a good omen. The address is 2313 which does add up to 9, not almost 9. I believe this was one of many small things written into the script to show Rocky's lack of education and intelligence.