Sponsored domains have a (generally nonprofit) sponsoring organization overseeing the domain to ensure it is operated in accordance with its charter in order to serve the needs of a particular well-defined segment of the Internet public (e.g., museums for the .museum domain). The registry (generally a commercial enterprise) operates the domain under contract with the sponsoring organization.
In contrast, an unsponsored domain is run directly by the (usually commercial) registry, and while it may have registration restrictions, its purpose for existing is to make money for the registry rather than to serve a particular community, though its marketing angle might involve trying to be of specific use to a particular community anyway.