What is the advantage of using REST instead of non-REST HTTP?

Simply put, REST means using HTTP the way it's meant to be.

Have a look at Roy Fielding's dissertation about REST. I think that every person that is doing web development should read it.

As a note, Roy Fielding is one of the key drivers behind the HTTP protocol, as well.

To name some of the advandages:

• Simple.
• You can make good use of HTTP cache and proxy server to help you handle high load.
• It helps you organize even a very complex application into simple resources.
• It makes it easy for new clients to use your application, even if you haven't designed it specifically for them (probably, because they weren't around when you created your app).