It's been nearly 35 years since I lived in Australia, but from what I've been able to tell I don't think the situation has changed that much. I think you would find more similarities than differences between the Australian and New Zealand cultures.
Both areas were originally settled by immigrants - the aborigines of Australia coming from Asia and the Maoris of New Zealand coming from the islands of the South Pacific. Both received large influxes of population from western Europe, although New Zealand's European settlers were not prisoners as was the case with Australia's penal colonies. In recent times, both countries have seen increasing numbers of immigrants from Asian countries.
New Zealand probably has more ethnic diversity and accepts it more readily than does Australia, particularly in urban areas. However, both countries recognize the importance of all the cultural heritages that are contributing to the continuing development of their societies.