Continuity mistake: Notice Marty's hair in the scene when he's playing "Johnny B. Goode." It's quite larger than when he says, some seconds later, "I guess that you're not ready for it yet." That's because when he says that, it's a recycled shot from the original BTTF, whereas most shots showing he's playing the guitar are newly made for BTTF2. However, Marty's hair is considerably longer in the new shots.
Revealing mistake: When Doc and Marty arrive in 2015 when the car is landing in the alley you can see the mechanism that is lowering the car. Just as the scene switches from the front of the car (right after the headlights go out) to the back of the car, look between the car and the building at the same level as the back bumper. There is something that is moving down with the car. Also, the vents in the back of the DeLorean are different.
Continuity mistake: In the scene where the 1985 Jennifer is in her 2015 home and says "I get married in the Chapel-O-Love?" right before she picks up the picture note the Chapel-O-Love picture and frame on the bookcase behind her. After she picks it up and drops it leaving it on the floor she looks for a place to hide. Later on when the old Lorraine and George are in the living room, take another look at the picture on the bookcase; the Chapel picture is set back up. You could argue Lorraine would have had enough time to set it back up. But look at the Chapel picture; it's inserted in the frame upside down. You can tell by looking at the frame stand behind it. Meaning someone physically removed the picture and put it that way.
Continuity mistake: If you look very closely, after Marty flips on the little girl's hover board and is laying on his stomach, when he goes to get up you can see that he's wearing regular lace style tennis shoes and not the auto-lace high tops that Doc gives him as soon as they get to 2015. You may need to pause it to see them.
Continuity mistake: In Biff's apartment in alternate 1985 while Biff and Lorraine are arguing he knocks her to the ground and both her shoes come off. After Biff punches Marty in the stomach it then switches again to show Lorraine's reaction. As it does this we see her on the floor but her shoes are back on her feet. She could have had enough time to put them back on possibly but it seems unlikely that she would have done so as she is laying in the same position as when she was knocked down.