New Zealand declined to join the Federation of Australia is the short answer.

Before federation in 1901 Australia was composed of separate colonies, essentially self-governing colonies from 1852 until 1901. New Zealand was also a self-governing colony in this period, and participated in the Federal Council of Australasia, a body that acted as a fore-runner to the Commonwealth of Australia, but with a slightly different composition. It met between 1885 and 1899.

The process of Federation in the late 19th century leading to referenda in 1898 (failing in Western Australia), and then successful referenda in 1899/1900. Australia then federated in 1901, becoming a Dominion of the British Empire.

New Zealand began as a colony administered from/as part of New South Wales, becoming a separate colony in 1841, and a self-governing colony in 1852. NZ declined to join the federation of Australia in 1901 and instead became, like Australia, a Dominion (and so effectively a nation) in 1907.

So that is why they are separate countries. I suppose the next logical question is why New Zealand declined to join the Commonwealth of Australia