Answer by Will Wister:
“Which” is preferred when you have a very limited number of known choices.
- Which arm is stronger?
- Which of your siblings do you like best?
“What” is preferred when you have a large number of choices or you don’t even know how many choices there are.
- What color do you like best?
- What is the best company that does X?
- What was the most impressive empire of all time?
Here are some references that back this up: