I'm a US native speaker. I don't think there's any difference. It's the usual English double vocabulary: "right now" is Anglo-Saxon-derived, everyday, less formal, less educated; "immediately" is Latin-derived, slightly more formal, slightly more educated.
In real life the intensity is conveyed by tone of voice, not choice of word.
In the command "Do X right now," "right" has the same sense as in "directly." X is the next thing you must do. Do not do anything else in between "now" and doing X. In "Do X immediately," the "im-" prefix means "not," and "-media-" means "in the middle" or "in between." So, it is the same idea: nothing should be in between "now" and "doing X."
In old British books, I've also heard the word "directly" used: "do it directly." This is the same idea once again, with "-rect-" being the same root as "right." I don't know enough about British culture to say whether it's still used or what shade of meaning it has.
They mean pretty much the same, but the urgency in my opinion depends mostly on the intonation.
"I need you to come inside, right now!"
"Get inside immediately!"
Both sentences, if said in a strict voice, let someone know that they better get inside really fast!
"Can you please come inside. I need you right now."
"Do I need to come immediately, or can it wait 5 minutes?"
In these sentences, there's still a feeling of urgency, but much less than in the first two sentences.