Statutory law and common law aren't usually examined to discover which one is better; most often they're viewed as two parts of a legal system that function well in tandem—one makes up for the gaps in the other and vice versa. Here's the short version of how they work:
Statutory law (sometimes just called statue law) refers to the body of laws that have passed through Parliament. Statute law can vary from state to state (e.g., different states might have different laws regarding the age one needs to be to get a driver's license, etc.). In order for something to become statutory federal law, it must be drafted into a bill (or an amendment to an existing law). It must pass through three readings in the House (in which questions are asked, changes are proposed, and votes are taken). If the bill is passed by the House, it goes to the Senate and passes through the same process. If passed by both the House and the Senate, it goes to the Governor-General, who gives it the Queen's seal of approval. State parliaments go through a similar process, though details may differ a little from state to state.
Common law, on the other hand, is set forth by the superior courts of Australia (the Supreme Court in each state, the High Court of Australia, and the Federal Court). It falls to the judges of these courts to resolve issues that are not covered by statutory laws, or in which the statutory laws are unclear or vague in the context of a certain case. The courts' decisions and interpretations of the law set a precedent that other courts refer to when making similar decisions in the future. All lower courts are obligated to follow the precedents set by the superior courts; only superior courts may overturn them.
Neither system by itself is perfect. For example, statutory laws are general rulings and decrees that by their nature can't take into account the particulars of a case or unusual circumstances. Although the long parliamentary process with discussion and amendments helps prevent the passage of laws that are unfair or ill-considered, unforeseen circumstances and changing situations require that the law be flexible enough to carry out justice in all situations. That's one reason common law is so helpful; it allows for the examination of individual circumstances (something we'd all want if we were on trial!) and helps clarify places where statutory law is vague.
It wouldn't be good to just have common law in place, though. Common law is set forth by a relatively small group of people: the superior judges. It wouldn't be a very democratic system if it let such a small body of officials determine the laws for all of Australia. Thus the debate among many elected representatives in the House and Senate help ensure the people of Australia have a strong voice in determining the laws that govern them.