 Re: Put away/Put back
Either is fine, and in many contexts they would mean the same thing. But as PronouncePro suggests "put back" indicates that they must go in the same place as they came from - which in some cases might not be where they are generally stored. EG :
- I found these books on David's desk. Do you think he'd mind if I had a look at them?
- No problem. he's not here today. But put them back on the desk this evening because I know he needs them tomorrow.
"Put them away" on the other hand assumes that the things are kept in a specific place, and the speaker wants them put in that place.
- Do you like my new shoes?
- Yes. But don't leave them in the living room. Put them away in the cupboard.
Here "back" is not possible, because the shoes have never been in the cupboard before. It's just the "right place" for them.
In many contexts though, both are possible.
- Where are the books I left on this table?
- Oh sorry - I didn't know you were still using them. I put them back/away in the cupboard.
"back" is possible because the speaker assumes they came from the cupboard and s/he is returning them; "away" is possible because it's the normal place where they are stored.
So : "put away" = tidy up. "put back" = return | 
Linear Mode
