My client from the US wants to have a website for Australian users. I am wondering if the web host really needs to be in Australia for this site. Does it really make any difference compared to having it hosted in the US?
Sort of. Two things to consider (although both are related):
Because the host will be on the other side of the planet, it will be relatively slow to respond versus hosting based locally. Since page speed directly correlates to positive conversion rates and a positive user experience hosting the site in the US may have a negative effect on their user experience and conversion rates.
Page speed is a ranking factor for Google. Sites that are slow will have their rankings affected. Keep in mind that fast sites do not receive a bonus and only the slowest sites are supposedly affected. But since their hosting will slow their site down if it is hosted in the U.S. this should be considered.
It won't make much difference in case you hosting has geographically distributed servers.
Still, you can take benefit of CDN to have goo page load time.
Take a note, "Only less than 20% of overall response time is spent downloading actual HTML document; rest over 80% of time is consumed downloading components associated on the page."