"Give me a knife, please."
Translation:Дайте мені ніж, будь ласка.
What is the difference between saying Дайте мені ніж, будь ласка instead of Дайте, будь ласка, ніж? Is one more polite than the other?
They are equally polite. The later is rather colloquial. In an everyday situation, it is often clear from the context that the person is asking for the knife for her/him self, therefore "мені" becomes superfluous and is omitted.