Why did the Prophet (peace be upon him) permit the Companion to have a nose made of gold, yet prohibit a woman from using hair extensions, even though both were out of necessity?

There is no contradiction between the prohibition of hair extensions and the permissibility of having a nose made of gold, because repairing the nose is a necessity. Arfajah ibn Asad was permitted to have a nose made of gold after his silver nose became putrid, whereas hair extensions are a forbidden deception and fraud, and not a necessity.