The main reason that Australia is more wealthy than New Guinea, according to Guns, Germs, and Steel by Jared Diamond, has to do with its colonization by Europeans. The indigenous peoples of Australia still live in poverty, and have a heritage of being oppressed by European settlers.
For Diamond, the reason that the Europeans who settled Australia developed such wealthy and powerful civilizations had to do with two accidents of geography. This first was that Europeans lived in an area where there were a plethora of domesticable plants and animals. This allowed for a neolithic agricultural revolution occurring relatively early in European history, and setting the stage for specialization of labor and the development of complex societies.
The second advantage that Eurasians had was living on a continent that was aligned on an east-west axis, allowing for easy trade and communication between groups living at the same latitude, who could exchange knowledge of agriculture and animal domestication. Native Australians and New Guineans had none of these advantages.