When I typed in the 10-character ISBN, it pulled up a book with a different cover, and a 13-digit ISBN.  the directions say that if all the info is EXACTLY the same, I can't list it using that listing.  A missing 13-ISBN is not the same as a registered real number.

Do I need to create a new listing?

No, if the 10 digit ISBN on your book matches the 10 digit ISBN in the listing then the ISBN matches, and you are fine on that point of information. There are lots of books that were printed with only a 10 digit ISBN, the 13 digit ISBN came later. (But because of the way that ISBNs are generated, there is always a way to generate a 13 digit ISBN from the 10 digit one, and that's what is in the listings, regardless of which ISBN is printed on the book.) Its also very common for a book to be printed with only the 13 digit ISBN but not the 10, which is also fine for posting. Covers do not need to match.

What Sara said   Unless you know how to read the 10 and 13 digit numbers, they do not appear to match one another.  If your book happened to have both listed and one matched PBS and one didn't...that could be an issue.  The cover picture does not have to match the PBS listing.  Only ISBN, title, author, and binding (did I miss one?).

And some trivia:

When you look at the two numbers together, you'll often see "978" at the beginning of the 13 digit ISBN.  ("979 is currently available for use, too.)  The last digit of both ISBNs is a "check sum" digit that follows a mathematical formula.  So if you drop the 978, the next 9 digits of both ISBNs should match, and the last digit won't match:

ISBN-13: 9780140437881
ISBN-10: 0140437886

Also, if you find an older book with a 9-digit "SBN", you can often add a zero to the front to get a valid ISBN.  This is okay to do per site rules.  Just be sure your book matches the listing.

What Patty said.

Also note that PBS lists both the ISBN-10 and ISBN-13 numbers for books that have ISBN numbers assigned to them. Either of these numbers can be used when posting a book. Older books published prior to about 1970 do not have ISBNs and will be listed here with short 3- to 6-digit numbers assigned by the site that will be displayed in the ISBN-13 field. Books published after about 1970 will generally have ISBNs assigned to them, and there is a simple (but outside the scope of this post) mathematical formula to convert ISBN-10 to ISBN-13 numbers, and vice versa. Also, many books published after 1970-ish have not been assigned an ISBN number and thus will be listed here using the PBS-assigned short numbers in the ISBN-13 field.

In the first few years when the SBN / ISBN numbering systems were being developed in the late 60's,  9-digit SBNs (Standard Book Number) were used, but these soon changed to the 10-digit ISBN (International Standard Book Number) system. As Patty mentioned, the 9-digit numbers can be converted to 10-digit numbers by adding a zero in front which does not affect the calculation of the check digit. The current ISBN-13 system came into common use in 2007, but some books published a little prior to this date also have ISBN-13 numbers.