No thanks is not more rude than no thank you, it is just more informal. Someone could interpret that as rude, if they are very touchy, but no thanks is definitely not rude to say. I use them both frequently. Cheers! :)

It is not rude at all. Both mean the same thing. :)

It depends on the way you say it. For example, imagine you meet pushy person, who tries to sell you something. You calmly say "No thanks" but he still insists. If you get irritated and rise the pitch of your voice, then it doesn't matter if you say "no thanks" "no thank you", you will sound rude as hell :)

Both are polite. "no thank you" is more formal, and therefore probably sounds more sincere

Saying 'No thanks' IS a polite way of saying 'I don't want to'. (싫어요. 하지만 [생각해 주셔서]고맙습니다) However, both 'No thanks' and 'No thank you' sound very formal without having REAL appreciation. If you want to be more polite, you might want to say 'No, I'm fine. Thank you, though'