Does the time difference between Chicago and Seoul change during the year?

Yes. Because of daylight saving time the gap moves between −15h and −14h.

When is the best time to call Seoul from Chicago?

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

Chicago is 14h behind Seoul. This gap is not fixed: daylight saving time moves it between −15h and −14h during the year.

Chicago

United States

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Seoul

South Korea

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

City	Time zone	UTC offset
Chicago · United States	America/Chicago	UTC−5
Seoul · South Korea	Asia/Seoul	UTC+9

What is the time difference between Chicago and Seoul?: Chicago is 14h behind Seoul. This gap is not fixed: daylight saving time moves it between −15h and −14h during the year.
Does the time difference between Chicago and Seoul change during the year?: Yes. Because of daylight saving time the gap moves between −15h and −14h.
When is the best time to call Seoul from Chicago?: There is no overlap between standard working days, so one side has to call early or late.