There were a number of questions after I reported that a new Google+ Local claim of a business in Plus (not the Dasbboard) generated instructions to add the rel=publisher rich snippet to your website. Many asked whether rel=publisher should replace the rel=author snippet. The answer is that you should do both.
Its not a matter of either or, its a matter of both and. You want to setup the rel=”publisher” to provide a context of identifying your business website to Google and especially giving you the chance to give them the categories that your business best fits, and then tying all of that back to your NAP information that Google has on file.
You want to setup your rel=”author” markup to recognize your contribution to the content as published by the business, but recognizing that your role as a human being is larger than just your position at that business. Maybe you also have a blog, and a hobby website. If you setup the rel=”authorship” formatting on all of those sites then your online identify or persona becomes clearer to Google as whole.
That said both are needed, just like getting a yellow page listing for your business and business cards for yourself are helpful so that people can find the business but also personally connect with you as a person.