Phrasal verbs

From Collins Dictionary:

put away
vb (tr, adverb)
1. to return (something) to the correct or proper place he put away his books
2. to save to put away money for the future
3. (Law) to lock up in a prison, mental institution, etc. they put him away for twenty years
4. (Cookery) to eat or drink, esp in large amounts
5. (Life Sciences & Allied Applications / Veterinary Science) to put to death, because of old age or illness the dog had to be put away

put back
vb (tr, adverb)
1. to return to its former place
2. to move to a later time or date the wedding was put back a fortnight
3. to delay or impede the progress of the strike put back production severely

As you can see, definition #1 for each is quite similar, whereas the other definitions are quite different from each other.

Distinguishing between definition #1 for put away and put back, the 'former place' does not always have to be same as 'the correct or proper place'.  It can be, but doesn't have to be.  When they are the same places, either phrasal verb could be used for definition #1.

He put his coat away in the closet.

He put his coat back in the closet.

He put his coat back on the sofa. (Not: away on...)

I hope this helps.

Shawn