Once a term known to only a few, "Refactoring" is now commonly tossed around the computer industry. I like to think that I'm partly responsible for this and hope it's improved some programmers lives and some business's bottom lines. (Important point, I'm not the father or the inventor of refactoring - just a documenter.)
However the term "refactoring" is often used when it's not appropriate. If somebody talks about a system being broken for a couple of days while they are refactoring, you can be pretty sure they are not refactoring. If someone talks about refactoring a document, then that's not refactoring. Both of these are restructuring.
I see refactoring as a very specific technique to do the more general activity of restructuring. Restructuring is any rearrangement of parts of a whole. It's a very general term that doesn't imply any particular way of doing the restructuring.
Refactoring is a very specific technique, founded on using small behavior-preserving transformations (themselves called refactorings). If you are doing refactoring your system should not be broken for more than a few minutes at a time, and I don't see how you do it on something that doesn't have a well defined behavior.
I realize I may be fighting a losing game here, but I do want to preserve the precision of the DefinitionOfRefactoring. There may be other good techniques for restructuring, but they are different. I'd like us to be clear about what we mean when we use this word.