Doing so does confer SEO benefits. Google is left in no doubt about your region of operation, even if your hosting is based overseas. So you will tend to be listed higher in searches by people in your own country.
But it's not like you can't get good local traffic without one. This site is a standard .com. And I have a couple of others without the specific country suffix. They all do pretty well in local searches. (Actually the main reason I didn't buy a .au name for this site is because I didn't really see this as a business in itself -- even though I do have a sole trader ABN.)
That said, I think that if you are an Australian-based business then you should have one. The main reason is that it is pretty much standard practice for Aussie businesses. It makes you look established and serious about what you do. So if you don't have one it might jar a little with some people who visit your site.
Still, you don't absolutely have to register one. And there are some big and well known businesses without them -- Australian Super and Telstra Bigpond for example.