Have had many years of experience at both usyd (did my BSc Adv Math and honours there) and ANU (have been doing my Phd in pure math here for the last few years, and have taught and observed various undergrad courses).
They are both strong programs and turn out many fine graduates to both research and industry. For what it is worth, I think the undergraduate pure math courses at ANU are more challenging than those at usyd, but of course some of the later courses are available at one uni that aren't at the other, based on the specialties of lecturers.
From my experience these are two of the highest quality undergrad programs in the country when it comes to producing research quality students who go on to get postgrad positions at strong universities (the US being the typical destination).
Of course, this might not be your goal at all, or it might be far too early in your education to even know. That is why it is not so important that you choose the "best" program (if such a thing exists), and rather just choose a strong program with lots of flexibility in courses so you can explore your passions and see what you actually want to do. Keep in mind you can also transfer from one to another for your honours year, which several unsw students did to usyd in my cohort.
I have interacted less with people in applied math and stats and as such am less qualified to make claims about the comparative quality of those departments.
Note: I do not think that unsw is a bad university at all, but from memory I was not a fan of their structuring of the pure undergrad courses. Some big topics in analysis were lumped together and consequently given a more perfunctory treatment that my comparison point of usyd. Perhaps a matter of taste, but anyway, this is why it is important you talk to several people about this. And if you are a sufficiently driven and motivated student, you shouldn't be too constrained by the courses you take anyway.
