The best time to visit New Zealand really depends on your own preference - people come to NZ all year round as there is always something to do. Temperature varies across both Islands but is relatively moderate with no lasting extremes. There can be quite a lot of rain, but equally sunshine hours are plentiful. Bad weather rarely lasts more than a day or two and quite often you can experience sun, rain, wind and even snow in one day. Kiwis tend to layer up for the cool or wet hours & shrug off their extra clothes when the sun comes out, as it inevitably does. Always carry sunglasses and sun lotion plus a rain jacket. When the sun comes out, the heat will dry you off very fast!
Most people travel to New Zealand from December to February, the NZ summer, but the busiest times are during local annual summer holidays - generally from Christmas through January. But New Zealand is large enough with so few people that you'll inevitably be able to find a quiet spot, if that's what you're after.
Spring (Sept-Nov) is full of colour but a bit wet. It's the best time for white water rafting as the river levels are high. Daylight saving starts at the end of September, so the days are longer. The last snow falls in the higher Alps. Summer (Dec-Feb) reveals how fierce the NZ sun can be! Slip, slop, slap & wrap is the slogan: slip into a shirt, slop on some sunscreen, slap on a hat and wrap on a pair of sunglasses because you will need protection. Summer has the most sunshine hours & the least rainfall, but the possibility of a tropical storm is always there. Autumn (Mar-May) is beautiful with autumn colours appearing, nights cooling (always welcome after summer) but with the heat still present. Daylight saving stops early April, so the days are shorter. If camping is your thing, some campgrounds close after Easter. Winter (Jun-Aug) is obviously colder, so select accommodation carefully to get heated, insulated rooms. Skiing is a major winter activity, so is diving (in the North) and fishing. Walking during the day is especially rewarding as the air is particularly fresh & the sun is still very warm when it's out.
The school summer holidays (mid December to end January) are the very busiest times and the only times you really need to be careful about booking accommodation ahead in the most popular areas. Public holidays fall on New Year's Day & the day after, Good Friday, Easter Monday, Christmas Day & Boxing Day. Other holidays are Waitangi Day on Feb 6 (celebrating the day the treaty founding modern New Zealand was signed between the British & the Maori), ANZAC Day (remembering all Australians & New Zealanders killed in war) on April 25, Queen's Birthday on the first Monday in June & Labour Day (commemorating the successful struggle for an 8 hour working day) on the fourth Monday in October.
Because there are many microclimates in New Zealand, it is very difficult to simply define the differences in climate between the North & South Islands. Broadly speaking, the North Island is warmer & it certainly never gets as cold as the South. However, Nelson at the top of the South Island has the most daylight hours of anywhere in NZ. During Summer, there is never a great deal to separate the Islands, each having their warm & cooler spots. During Winter, the South Island is generally colder, but that cold reaches half way up the North Island as well. One of the best skiing fields is just 4.5 hours drive south of Auckland! North of Auckland, the temperature is inevitably in double figures, year round.