There's barely any similarity apart from what you mentioned.
Karate kid: kid moves because her mum got a new job, kid has basically 0 karate experience gets bullied for chatting up the bullies girlfriend and mr Myagi saves him and trains him for a tournament.
NRNS: kid moves because his karate teacher dad is getting hassled by the mafia, kid has good martial arts knowledge not an expert but pre training Jason could easily have beaten pre training daniel. He gets bullied for sticking up for his friend and because of lies being spread about him. Then the spirit of Bruce lee trains him not to compete and get revenge but simply to be a better martial artist and only fights when his girl is hurt.
So really this is barely any similarity apart being a martial art movie with a kid moving getting bullied then training which like has been said is hardly an original concept in the first place
The similarities are glaring. Moving across country, conflict with bullies, rivalry over a girl, argument with the girl, makes up with the girl, Chinese mentor and there's even the money shot special movie (the crane/the backwards somersault kick). I love this film, but come on. The only difference between this film and the Karate Kid, is that Bruce Lee didn't get his car washed or his fence painted.