Where the two terms come from
In retail, white label traditionally described a generic product made by one manufacturer and sold by several stores, each with its own label. Private label described goods made for one retailer alone, sometimes to that retailer’s own specification. The distinction was about exclusivity and customization rather than about whose logo appeared on the package.
Software vendors borrowed both phrases. Some use private label to suggest a more customized or exclusive arrangement, and a few use it for software installed on the reseller’s own servers. Most use the two interchangeably to mean a product sold under the buyer’s brand. Neither term has a legal or industry-standard definition in accounting software, so a vendor’s wording tells you little on its own.