"Right now" is used more in situations where you are either about to do the thing or you are actively doing the thing. "Right away" means you will do the thing as soon as possible, but doesn't mean that you are actively doing the thing. The two phrases are interchangeable when you mean to say that you will do the thing next/as soon as possible.
They do have slightly different nuances, but let me think of how to explain....
I suppose it just depends on the person, but "right away" is a bit more polite when asking someone to do something as soon as possible. Asking someone to do something "right now" is a bit more accusatory, as in you did something wrong and need to fix it. Ex: "I need you to fix this report right now." It also gives a request a more serious edge. "I need you to take this there right now," is emphatic and almost rude. It might be said when the person making the request is frustrated. "Right away" is more gentle/polite when you ask someone to do something.
Did that make sense ?