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

Yes. Because of daylight saving time the gap moves between −14h and −13h.

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

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

New York is 13h behind Seoul. This gap is not fixed: daylight saving time moves it between −14h and −13h during the year.

New York

United States

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Seoul

South Korea

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

City	Time zone	UTC offset
New York · United States	America/New_York	UTC−4
Seoul · South Korea	Asia/Seoul	UTC+9

What is the time difference between New York and Seoul?: New York is 13h behind Seoul. This gap is not fixed: daylight saving time moves it between −14h and −13h during the year.
Does the time difference between New York and Seoul change during the year?: Yes. Because of daylight saving time the gap moves between −14h and −13h.
When is the best time to call Seoul from New York?: There is no overlap between standard working days, so one side has to call early or late.