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.
When 速食面/ 方便面 (instant noodle) was invented, you needed to put the noodle into boiling water and cook it for 3 minutes. That requires using of a stove (it is still the majority type today)
泡面 is a later version of instant noodle. All you need to do is pour boiled water into the foam cup (the foam cup is also the package) and cover it up for three minutes. Since most family have electric hot water dispenser, you don't need to use the stove to cook instant noodle anymore. The noodle is also called 'cup-noodle' (杯面) and all cup-noodle are '泡麵' ;
There are some 泡麵 come in regular package instead of foam cup package. you have to use your own bowl
'泡' in '泡面' means 'infuse', not 'bubble'
If Taiwan people call all instant noodles including the ones that needed cooking as '泡面', it may had been a generalization practice. Just like most Hong Kong people call all instant noodles 公仔麵. BTW, The official name for instant noodles in Hong Kong is 即食麫. In Guangdong , also called 即食麫.
Update:
There's a new style of Japanese ramen noodle called 泡系拉麵 (foam series ramen) which the soup of the ramen is thickened and foamed. If this style became popular in Taiwan, the term '泡麵'(foam noodle) may be confused with '泡麵'(infuse noodle)
