I recently bought a book at the thrift store and I went to enter the ISBN to add it to my TBR list.  I entered the10 digit ISBN from the back of the book and it was not found in the system.  So I entered the 13 digit ISBN from the back and it wasn't found in the system either.  I look on the copyright page inside the book and it lists a different 10 digit ISBN than the back of the book.  I enter the one from the copyright page and it shows up in the PBS system just fine.  Why would the number be listed differently on the back versus inside the book?  I have never encountered this before.

The ISBN-10 and ISBN-13 on the back of the book are really the same number. The ISBN-13 just starts with 978 and has a different final digit. If you know one of these numbers you can calculate the other. So if one of them is not in the system, you don't need to try the other one, it's guaranteed not to be there either.

As to why there are different ISBN numbers on the copyright page, sometimes the original numbers from the original edition are left on the copyright page for future editions. You can post your book using these numbers only if the book type matches the listing. If those numbers from the copyright page belong to a hardcover and you have a paperback, then you cannot use them. In that case, if the ISBN on the back of the book doesn't work, the only thing you can do is use the Post a Book Without an ISBN page.

I think when you search an ISBN and it doesn't come up like that it prompts the system to find the book and ad it.  So if you check again in a few days it might be there. 

Soemtimes (rarely) there is just a misprint of the ISBN. I have a book or two with a ISBN sticker covering up the misprinted ISBN sticker.

Sometimes an edition will come out with a different number so the publisher can track where the books were sold to.  So a special edition with a different number will be sold to a large chain of gift stores or a book club.  And the standard number going to bookstores.