Both movies are about two teenagers who move to a new place and who don't fit in.

Also in Karate Kid, Daniel wears a red nylon jacket the same as James Dean.

*His ex girlfriend was called Judy- which was the name of James Dean's character's girlfriend.

* During the fight on the beach, Johnny says to Daniel "Hey Toro" which also happens on RWAC.

* There's one other scene where Daniel says something to his motherr (can't remember what exactly) and it sounds very similar to something said in RWAC.

I think there are more similarites, but I just can't remember them right now.