The main distinction is that the "/" search is done by Lynx, but the "s" is done by the server the document came from. Since the second one is done by the server, the document has to be set up for that type of searching and the server has to be configured to accept it. This is usually done through CGI, but the search may also be submitted to a gopher server or a WAIS server.
A second distinction is that the "s" search does not search the current document, but instead searches the database set up by the document and the server. The "/" search is used to locate text within the current document.
Would you like to view the key bindings for the version of Lynx you are currently using? This won't work if you are not using Lynx, by the way.