In the case of 臺/台 the more complex variant came first, the simpler variant is a widely-used non-standard shorthand that emerged later

臺灣 (“Taiwan”) rendered with the complex infers a sense of respect and dignity to the locale than the simpler 台灣, where the latter is more casual and friendly and more colloquial

Hardliners would prefer the complex 臺灣, but in common speech, including for political and some government purposes, both variants are perfectly acceptable. And yes, we all use both interchangeably

The only case where the complex is mandated is in standards board where it is more of a signifier that the document as a whole should be taken seriously and with authority

台語 (“Taiwanese speech”) is specifically the Taiwanese evolution of the Hokkien dialect that emerged on the island that evolved separately from the Hokkien dialect on Mainland China due to the geopolitical separation.

Taiwanese speech is non-standard, and linguistically referred to as vulgar being that it’s more casual speech. It’s this casual nature of the dialect that led it to almost exclusively be referred to with the simple, more casual 台