Web to Case doesn't use REST or SOAP API; it's a simple HTML form with a POST submit. You can paste the code verbatim into a Visualforce page, expose it on your Site, and it will be behave as any other form; you can capture cases automatically.
If you want to use the REST or SOAP API, which you can, you wouldn't be using Web to Case, which is a simple POST to the server. Web to Case would take just minutes to set up on a Site (minus any styling you need to make it your own design). Using web services, you could be manually creating the records via the API.
If you chose to go with web services, you may as well just create a Visualforce page and have it designed like any other Visualforce page; there's no need to leverage the API just to submit a case through Sites. Note that you can only "create" cases, not update them, using the Sites technology.