The have the same meaning. "at once" may be a little stronger but they both refer to the same time - straightaway, right now, immediately, etc. Also, "at once" can mean "at the same time": I'm doing many things at once.

There is very little difference in meaning between the two. If your boss says "I need this done right away," it means the same as "I need this done at once." That said, I'd consider "at once" to be slightly more forceful and immediate. The expectation is that without any delay the hearer would stop what they are doing and change to do what their boss demanded. If he said "right away," maybe a brief delay would be acceptable.