Archived listings are ones that you have sold or leased. Published/Unpublished listings will show or not show on the website. This is if you are still working on the listing. It was introduced for our new integration with Buildout. So, if you are syndicating a listing from Buildout to QuantumListing, there is an intermediary step on QL of publishing it before it is visible on the site.
If a listing is leased or sold, you only need to archive it, not unpublish it. And, hopefully you'll also add it to our comparable database!