Are you thinking of publishing a paperback version on Amazon? They offer a free ISBN but is it really free? With it, comes expectations and restrictions. There are 3 ISBN options:
Amazon prints paperback in their CreateSpace. By using their FREE ISBN, they will become your distributors for this book’s edition and ISBN. Their imprint will be shown on your book’s detail page.
By purchasing your own ISBN, you can have your own imprint (brand) and also choose your own distributor.