Both Australia and New Zealand have a shared history as being a part of the former British Empire, having been colonized in the 18th century. But they both have prior indigenous populations that predate European settlement that are very different. Indigenous Australians are called, for lack of a better term, Aboriginals. There is no single tribe or lineage that they describe themselves as. Instead, they are composed of up to 250 different groups spread across the island country (this is partially why there is no more distinct term for them as a whole). Aboriginals have been in Australia for 40,000-50,000 years having crossed onto the current island from southeast Asia and the island of New Guinea either on foot via a former land bridge or by boat. They probably represent the single longest continual distinct culture still intact on the planet. Native New Zealanders, in contrast, are a distinct culture that are called Maori. They are descended from a different group of people called the Polynesians who made their way across the various islands of the south Pacific by boat over the centuries. The Polynesian Maori did not arrive on New Zealand until around 1300 AD. So both the origins and the length of time of existence are the two big differences between the indigenous populations.
The immigrant cultures of both nations is more similar than their indigenous ones, due to the fact that white European settlement on both originated with England. Most of the populations of both nations are predominantly white of English origin, but other European nationalities like Dutch, Italian, and Polish are present as well. In more recent decades, non-white European immigration has increased, particularly from Asia. New Zealand probably has a slightly more varied immigrant mix, but not by a large amount. One feature that might distinguish the immigrant populations would be the purpose of immigration. Part (but not all) of Australia was originally founded as a penal colony, and other colonists came to the territory to farm the land and raise livestock. New Zealand, on the other hand, is a tiny land mass in comparison and has much less arable land for farming. So many colonists coming to New Zealand were there for ocean commerce and whaling opportunities, thus leading to a more itinerant population.