What are some down sides of doing PhD in an Australian university than that of the USA?

The other respondents have covered it well. I’ll still answer redundantly.

• No coursework; so you can emerge with gaps in your knowledge about the discipline. I know I did.
• Not necessarily much of a seminar culture (may vary by faculty); so much less opportunity to refine your ideas against your peers.
• Much less networking opportunities, as it is a smaller country. Which means more corpses to step on if you’re going to end up with an academic position.
• One overseas trip if you’re lucky, when you’ve got to fit in any networking at conferences.
• No viva examination at the end of the PhD; so no sense of ceremony or moment, and no opportunity to defend your ideas.
• PhDs were introduced in Australia in 1948. Australia was at the time a lickspittle nowheresville colony as far as everyone was concerned (particularly academics): they deemed that noone in Australia was worthy of examining theses, and shipping candidates to Pomgolia for their examinations was not cost-efficient. So they had candidates submit for written assessment alone. And to this day, one of the external candidates has to be overseas.
• Pomgolia. Memorably defined once by People (Australian magazine) as “a small country to the south of Scotland”.