Apple Authorised Resellers are just what the name implies. These retailers are authorised by Apple to sell both new and renewed/refurbished Apple devices - the latter having been through Apple's official refurbishment processes.
Apple's own renewed devices are certified by Apple to be the equivalent of "new" and have an associated Apple limited warranty; these devices will have a new case and new battery. Apple refurbished devices are only available from official Apple Stores or Apple Authorised Resellers.
Not all independent retailers are Apple Authorised Resellers. Refurbished devices purchased from many independent retailers have been independently refurbished by a third-party, may not use OEM parts - and may be of questionable quality. There will be no Apple-supported warranty.
Caveat Emptor (Buyer Beware)