If you want to be a pilot in ANY of the 3 services, you need a Science based degree. End of story. If you wish to be in Infantry, Armour, Artillery, Combat Engineers etc, I don't think it matters.
In fact, for ALL the other Corps, as far as I'm aware, you don't "need" any degree per se to be an officer, though it certainly helps. It is also possible to accept an offer to go to RMC Duntroon and train as an officer, if you are a private soldier or NCO/SNCO as well, demonstrating that you do not "need" a degree.
However as a Commissioned officer, you WILL need a Degree to get promoted to the higher ranks. To achieve rank beyond Major, I believe you need a "Masters" Degree, though don't quote me on that. It's a LONG time since I saw the actual requirements. The ability to speak a foreign language fluently was ALSO a requirement...
It is also possible to serve as a Digger, NCO, SNCO and finally Warrant Officer and subsequently be promoted to Commissioned officer rank, through an internal Army scheme. However I believe Captain is as far as you can go that way, without having been Commissioned through RMC Duntroon.
As far as ADFA goes, I understand you are fairly restricted in the types of degree's you can get through this scheme, though the University of NSW actually provides you with the degree so at least it's worth something, outside ADF.
As Tod mentioned every state in Australia has "University Regiments". These are reserve formations which you join should you wish to become a reserve officer.
Basically you do a series of "modules" at RMC Duntroon (over a 2 year period, I think it's 4x modules of 4 weeks each or 8x module of 2 weeks each, something like that anyway) at your particular UR (I am only familiar with QUR - Queensland University Regiment, having "played enemy" on a couple of exercises for them) after completing basic training the SAME as every other Dig, as an officer cadet.
At the completion of this you are commissioned as a Second Lieutenant in Army and attached to a particular unit. You are not qualified to command a platoon at that point however, you need to work for a while, completel in-service training etc and become qualified for promotion to full Lieutenant for that.
Hope this helps...