Well, you don't actually use "yet" with simple tense. So, "The director hasn't come yet" is true. As to your question about when to use the sentences, you say "The director hasn't come yet" if you are still waiting for him (process is going on). And you say "The director didn't come" if everything has already ended (you waited for him, he didn't come).

For some speakers of American English, there is no difference, but the first sentence would be seen as more 'correct'. Outside of North America, the second sentence would be seen as incorrect. This is because 'yet' refers to an unfinished period of time, and when you have an unfinished period of time, you have to use the present perfect. The past simple should only be used with reference to finished time periods. As Zaur explained, 'The director hasn't come yet' means that you are still waiting for him (or her), so this describes the present situation. 'The director didn't come' would be used with reference to a 'closed' time period. For example, 'How did the meeting go last week?' 'Well, the director didn't come because he had another meeting that day, so his deputy came instead.'

Adding to Zaur, it would be considered incorrect to use "yet" with the past simple in the UK (though I am sure it happens!). However, I think that it is more acceptable in the USA, though the present perfect alternative is also common there. If any US teachers or tutors could confirm or comment on this too, that would be useful.