Ahh the old visa/permit question.

There's a very simple way to know what you need:

Visa - lets you in
Permit - lets you stay

If you are outside New Zealand, you will need to apply for a work visa, and then when you land you will be granted a work permit by immigration. If you are within New Zealand, you apply for a permit as you are already within the border.

Whichever way you apply (initially a visa in your case) you require a permit. Even with a visa, your permit can still be declined upon arrival in NZ, so you can be sent packing! Don't worry though, that only happens to people with fake documents or undisclosed criminal histories etc!

As immigration like to say - 'You must have a valid permit for the entire length of your stay within New Zealand'.