Does the time difference between Auckland and New York change during the year?

Yes. Because of daylight saving time the gap moves between +16h and +18h.

When is the best time to call New York from Auckland?

There is no overlap between standard working days, so one side has to call early or late.

Auckland is 16h ahead of New York. This gap is not fixed: daylight saving time moves it between +16h and +18h during the year.

Auckland

New Zealand

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New York

United States

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A 9-to-5 working day in Auckland does not overlap one in New York at all — somebody has to take the call outside office hours.

City	Time zone	UTC offset
Auckland · New Zealand	Pacific/Auckland	UTC+12
New York · United States	America/New_York	UTC−4

What is the time difference between Auckland and New York?: Auckland is 16h ahead of New York. This gap is not fixed: daylight saving time moves it between +16h and +18h during the year.
Does the time difference between Auckland and New York change during the year?: Yes. Because of daylight saving time the gap moves between +16h and +18h.
When is the best time to call New York from Auckland?: There is no overlap between standard working days, so one side has to call early or late.