When Daniel and Jessica are hanging off the side of the cliff being held hostage by Mike and his gang. Snake tells Daniel to give him the tree and Jessica, when telling Daniel not to, calls him "David" instead of "Daniel".
When Miyagi comes into the shop as Mike is fighting with Daniel, Mike tells Snake to "get him". After Miyagi drops Snake, Mike then points to Dennis, sort of waving him to go after Miyagi next. We see Dennis start toward Miyagi but in the next frame, Mike is the one going toward Miyagi, saying "come on old man".
This is supposed to take place 1 year after the events of the first Karate Kid. In the first film we see him turn 16 (Mr. Miyagi gives him a car for his birthday). So one year later he would be 17, yet he was supposed to go to college. Did Daniel graduate High School early? Was he in an accelerated class? Also, if he's still under 18 (which is why he can defend his title in the "All Valley Under 18 Tournament") How did he secure a lease with the real estate agent for the bonsai store? Are people under 18 years of age allowed to enter into legally binding contracts?
Towards the end of the movie Daniel tries to heal his arm using Miyagi's thick light green fizzy substance. In the next scene Miyagi knocks on Daniel's room's door which startles Daniel and knocks down the container with the liquid in it which now simply looks like water without any sign of the green stuff which was there just a few minutes earlier.
When Mr. Miyagi fights Terry Silver and the gang at the Cobra Kai dojo, he throws Barnes into the wall and temporarily knocks him out. Barnes falls face down with his head pointing away from the front door. In the next shot, they show Barnes lying on his back on the mat and his head is pointing toward the door.
When Miyagi sends Barnes flying through the Cobra-Kai door into the dojo, if you look closely, you can see some sort of table (with a blue tinge to it) that Barnes used to jump off of and into the door from.
When Mr. Miyagi fights Mike Barnes, John Kreese and Terry Silver, Kreese lunges at Miyagi, but Miyagi blocks Kreese's punch, ducks down, kicks Kreese in the face and trip Kreese. As Kreese falls to the ground, you can see something small, rectangular and shiny fall out of his shirt (most likely a pocket protector or name tag). In the next shot the thing is no longer on the ground and has completely disappeared.
When Terry Silver and Daniel are in the Cobra Kai dojo for the first time together and Daniel is attempting to sweep the "knee" of the wooden dummy, the wood is solid and he cannot break it. As Terry Silver prepares to demonstrate the move, a break in the wood of the first leg is plainly visible. Sure enough, that's where his foot makes contact and the wood falls apart. A break in the torso, where he hits it, is similarly visible.
When Barnes kicks Daniel in the groin in the match, the referee warns Barnes that one more illegal blow and he will be disqualified. Shortly thereafter Barnes "sucker-punches" Daniel in the back of his head while walking back to his position. This should have been a disqualification for that action alone along with the warning he had previously received, but instead the referee gives him the same warning that any more illegal contact will disqualify him.
When Miyagi rescues Daniel from Silver and Kreese in the Cobra Kai Dojo, Miyagi throws Mike Barnes into the light switches, which causes the lights to turn off in the room. Looking closely, you can see that the light switches don't actually change positions.
In the beginning when Kreese goes to his now-closed Cobra Kai dojo, he looks at a newspaper clipping about Daniel winning the All-Valley Tournament. You can see the text of the story under the by-line and it has nothing to do with the All-Valley. (Also, the caption of the photo refers to "Mr. Miyagi"--did he not want to give the reporter or photographer his first name?)
Towards the end, in the training sequence of both Daniel LaRusso and Mike Barnes, the first two clips shown of Mike Barnes, Terry Silver, and John Kreese (where Mike is breaking bricks with his hands) are the same exact clips, just played twice. The second clip of Mike is about half-a-second longer than the first.