In the accepted answer, what's so bad about the "non-restful" approach the author describes? Who cares if I use GET/DELETE/PUT/POST vs putting the "verb" in a query string argument or the name the file that's being called?
RESTful, as opposed to REST, seems to be clear about the use of HTTP and all of its verbs. It is not RESTful if you do it any other way just by definition.
REST itself does not concern itself with the protocol you use. Theoretically you could build a REST protocol over HTTP that does not use HTTP verbs, using it as only a transport layer. Though I'm not sure what you would gain by implementing a REST protocol over another REST protocol.
Pragmatically, if you choose to use HTTP, you need to think about the infrastructure that exists. Each verb comes with a very specific meaning, and systems like proxies and other software that come between you and the client will act in different ways depending on which verb you choose. To meet the goals outlined in the REST dissertation, you need to be aware of those behaviours.
