You use an archival-quality binder for the same reason you'd use other archival-quality materials. You don't want low-quality materials to outgas or to leach acid or other unwanted byproducts into the box that is holding the materials you want to archive.
The standard for storing and displaying photographic materials is the Photographic Activity Test (PAT).
Gaylord Archival's page that describes the specifications for their products includes the following notice:
Most of our storage materials have been submitted and passed the test. The absence of the logo does not indicate that a product has failed the P.A.T., as some products have not yet been tested.
You don't know the provenance of the binders you already have. You don't know the materials they were made of, or how they'll perform under the conditions your photos will be stored under. Yes, the sleeves and any materials that come directly in contact should be the highest priority, but over time as your budget allows, why not get archival binders and boxes too?
Against the cost of good quality archival materials, consider the time and expense involved with trying to restore photos which have been damaged by improper storage.
From Caring for Records at the Georgia State Archives: