It's been awhile since I've learnt this so I might get a few things mixed up.
Australia is a Constitutional Monarchy, meaning the Queen is officially the head of state/ruler but the ability to make and pass laws belongs to Parliament. The Monarchy part comes from our English history where kings and queens like Henry VIII and Elizabeth I were the rulers and law makers. At some point in history they decided that one person shouldn't have all the power so now that power is given to the government/parliament, but they still need final approval from the queen.
So basically, the government wants to make a change to legislation but they have to get the okay from the ruler of the country, aka The Queen. A lot of countries belong to the Commonwealth so obviously she can't be there for everyone so they/we set up a representative for her. The voice of the Monarchy in Australia. That is the Governor General. He gives the okay/declines laws the government want to pass in Australia. He has to remain neutral to again, avoid tyranny. Around 1980 or so, the government wrote into the legislation that Great Britian could no longer interfere with the way we run our country so officially, the queen legally can not step in and the GG position now is a last resort for bad laws making their way through. So if there is corruption all through the parliament, GG saves the people from giving up their souls since he's always supposed to think about the citizens best interests first. He is also the only one who has the ability to dissolve parliament. Lots of power but he is not allowed to use it lightly. Get this though, the Prime Minister chooses the Governor General. I think in approving a new GG, that is the only time Britian has any say here.
To answer your question, the Governor General is the representative of the Queen in Australia, so they're basically the same person. She is the head of state in title mostly, but Tony is the head of government/leader of the government which is the ruling party/governing party and "leader" of the nation. A bit like a CEO/manager thing. CEO (queen/gg) lets the manager do everything and run the business until he stuffs up royally and then the CEO steps in and fires him/lectures him, etc.
Australia is run strangely. We have a parliament which is British, but the American seperation of powers. If we went Republic, we'd pretty much stay the same but remove Peter Cosgrove.
www.peo.gov.au/learning/fact-sheets.html is an awesome site that teaches you about how we run as a country and exactly what the GG can and can't do.