Every listing on this site is one or the other, and the card tells you which: an agency listing carries the house name under the provider's own. Neither is better. They are different in ways worth knowing before you start.
An agency handles bookings, advertising and often scheduling. You contact the house rather than the provider; the house confirms, arranges the logistics, and takes a share.
The practical consequences are mostly good ones. Agencies answer faster, because answering is somebody's job rather than something squeezed between other things. They are more reliable about confirmations and cancellations. They can usually offer an alternative if your first choice is unavailable. And screening is generally handled once by the house rather than repeated with each provider.
The trade-off is distance. You are dealing with an intermediary, which means less back-and-forth beforehand and a booking that is arranged rather than agreed.
An independent runs her own listing and answers her own messages. Response times vary a great deal — she is doing this around everything else in her life — and confirmations are looser.
What you get in exchange is continuity. You are talking to the person you will be meeting, from the first message onwards, and everything you agree is agreed with her directly. For people who see the same provider repeatedly, that is usually the deciding factor.
Independents also screen personally, which tends to mean more thoroughly and with more judgement applied.
On every city page the agency listings carry a gold diamond and the house name beneath the provider's own name. Independents carry their city instead. You can also filter to one or the other from the search box.
Both kinds are hand-checked before they go up. Neither gets preference in the ordering — the only thing that moves a listing up the page is the VIP or Promoted tier, and those are labelled on the card so you always know when placement was paid for.