They have close to the same meaning, but they are used in different situations. "Right away" is often used in more formal situations and expresses a greater sense of urgency. Examples: (in a restaurant) "Please bring me my check." "Yes, sir. Right away." (a boss and an employee at work) "Did you do the reports?" "No, sir. I'll get started right away." (two friends) "What are you doing right now?" "I'm at home watching TV. You?" (a mother and daughter) "Do you do the dishes?" "No, not yet. Do I have to do them right now?"

there is a slight difference between them, "right away" focus on whats going to happen next; while "right now" more likely emphasis something happen by the time. To make it sound easier, the content "right away" held is a bit like running water and "right now" sort of like still water:)