When did 方便面 become 泡面 and why? I've noticed people avoid the word 方便面 these days, instead opting for the word 泡面. I'm not sure exactly when this switchover occurred but it does seem very pervasive at the moment. 



KEY does define 泡面 as:


  {TW usage} instant noodles




Is it just that 泡面 sounds fancier that it has cotton on?
I think, your intuition is right. 泡面 sounds fancier, and people prefer fancier names unconsciously.

方便面 sounds cheap; they were just instant noodles for filling one‘s empty stomach. The name reminds the way how migrant rural workers eat 方便面, which is very common to see in train stations of big cities.

泡面 sounds as elegant as 泡茶 (make tea); because usually it takes time to 泡 (verb) something and then to enjoy what has been 泡-ed.