The origins of the name "Australia" (in a way at least) predate those of "New Holland", because Medieval cartographers already speculated on the existence of a terra australis (Latin for 'Southern Land') long before Australia itself was discovered and often added it to their maps. The idea being, that God favored balance so that if he created continents in the Northern Hemisphere, he then must have also created (as of then undiscovered) southern continents. The Dutch themselves also used a calque of this term (Zuydland = Southland) as can be seen in this map, which (by the way) doubles as an explanation for why the Dutch never settled Australia as most of this particular map is explicitly marked as consisting of "extremely scorced land".

So when the Dutch explorer Abel Tasman named what is now Australia "New Holland" in 1644 (nearly forty years after Australia had been discovered by Willem Janszoon) maps were still being published with a terra australis, that is, to the Southeast and/or Southwest of New Holland. In fact, many map titles explicitly state that New Hollandis either in or part of the largely undiscovered Zuydland. There of course wasn't much more to discover in Oceania in terms of large landmasses, which is why the British, when they sought to rename New Holland, supposed that "Australia" should be used instead of "New Holland" as the probability of discovering another southern landmass equal in size to Australia, was (at that point in time, 1814) nonexistent.

In addition to this, after the discovery that Australia was the ´final´ continent to be found on earth, the unspoken opinion among some British colonialists was that a landmass of such importance oughtn't be named after a (especially at that time) an exceedingly small and somewhat unimportant European country. The explorer Matthew Flinders implied as much, stating that among Africa, America, Europa and Asia ... "New Holland" tends to stand out; which I personally (as a Dutchman) cannot argue with.

Another more practical but no less important factor was the establishment of New South Wales in 1788, which was initially seen as being located in New Holland; but at its greatest extent would encompass 2/3 of the Australian landmass + New Zealand. Its prominence resulted in "New Holland" being increasingly relegated to mean Western Australia, rather than the whole of the continent. During the 19th century (Victoria, Queensland, South Australia) several British colonies had been established or carved from New South Wales and calls were made to use the term "Australia" to refer to the continent on which they were located. One of the most influential and vocal proponents, the English naturalist Joseph Banks, noted that "Australia" would provide the continent with a more neutral name (ie. not one based on European polities) and one of greater antiquity and thus prestige. Eventually (though "New Holland" would continue to linger on for much of the 19th century) this won out.

In contrast, New Zealand was not renamed. Even though the British could have renamed it. There was historical precedent: certain Dutch colonies (ie. New Amsterdam > New York) had been renamed in the past. The reason they did not do this, most lies in the fact that it simply didn't warrant the effort.

Australia was a continent, whereas New Zealand was a 'mere' island. With "Australia" the British could lay claim to medieval and early modern European nomenclatural precedent with all the imperial prestige that came with it, whereas with New Zealand they could not. Lastly, the first European settlers to arrive in New Zealand did so as a part of the New Zealand Company in 1825. This was a semi-private British endeavor which dominated the first and most defining decades of British settlement and further solidified the name of the islands as all settlers came as men and women of the New Zealand Company, which kept their Dutch/cartographical name.