Save Search is what you would do when you have entered search criteria from a brand new search, or made changes to an existing search you have open that you want to save.
Save Search As would be used when you open an existing search or template and want to use it for another client or saving another copy of search with different name or values.
Both methods give you the option of naming the search and making it your default template, as well as saving it for either yourself or a client.