Start could also be the beginning of a race, "I will meet you at the start" Start tends to be used to ignite something like an Engine, in a car race, they may say "start your engines" Turn on tends to be used more with something Electrical. "It is getting dark please turn on the light" Both are fine for computer but may be used in different contexts,, also consider you may be told to Restart your computer after installing an update or to Reboot it which means the same but more tech speak.

The only context I can think of where they would mean the same thing is "start the engine" or "turn on the engine", when talking about cars, motorbikes etc. "Turn on the lights", "turn on the tap", "turn on the heating". "Start the lights", "start the tap", or "start the heating" don't make sense because in these contexts, the words mean different things.