In practical use, there is no real difference.
Technically "twilight" occurs both at the end of the day and at the beginning. That is, after the sun sets and before it rises.
But most people only think of the day ending when they hear "twilight."
"Dusk," on the other hand is only at the end of the day. And it is technically the end, the darkest part, of twilight.
In real use, "twilight" sounds more poetic.
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