Darjeeling is under West Bengal state and Gangtok is the capital of Sikkim. It takes about 4.5 hours to reach one place from the other by car.
Actually you can start with either place. If you visit Darjeeling first, then you can avoid some parts of the bad roads to Gangtok. There are some major road repair and widening work going on at the National Highway NH31A that leads to Gangtok. And from Bagdogra or NJP if you head straight for Gangtok, there is no way to avoid those sections of the road.
Now consider the return. Darjeeling is nearer to NJP railway station or Bagdogra airport than Gangtok. The usual drive time from Darjeeling to Njp/Bagdogra is 3.5 hours compared to 4.5 hours from Gangtok. Additionally since major repair work is going on at the national highway NH31A, sometimes you can get huge traffic jams while returning from Gangtok. It can throw your timings out of control and can sometimes get quite risky while trying to catch a train or flight. So you must start early.