When I bring in new clients, should I only ask them for their business address, or should I also ask them for their residential address? And which of the two am I required to ask for by law in Australia?
The rules and regulation vary from country to country and from industry to industry.
For example if you are a financial services provider in the economically developed world you will need to collect enough information to be reasonably sure that you are not laundering money for criminal or terrorist organizations and some countries that your country may have sanctions against.
For many industries exporting to sanctioned countries may be illegal (whether there are such countries is a different discussion altogether). And so on and so forth.
As far as billing is concerned though you will need to collect information that will allow you to charge their account/credit card and provide sufficient verification to the bank issuing the card that the person paying for merchandise is actually the owner or an authorized user of the card.
Now what address will be required for you to collect business/home/other or all depends on what is necessary for you to be able to bill your customer and be reasonably sure that on a credit card you will not get a chargeback if someone used a stolen credit card. And also to make sure that your product is properly delivered if you have physical product to sell. Normally in the US you need 2 addresses Billing and Shipping to complete online transactions for example Australia may not be dissimilar in that respect.
I don't know anything about the legal issues that might apply in Australia, but from a business perspective this seems pretty straight forward. These people are your clients, not your friends, you should only be asking for business contact information, nothing personal. I have no problem giving my business contact info to clients and/or vendors, but if they tried to contact me at home I would be unhappy with that.