Zhuyin is colloquially known as bopomofo for the same reason that we call a latin keyboard a qwerty keyboard. Each key on a Zhuyin keyboard represents a unique sound in Chinese Mandarin and bopomofo are the first five letters starting where the Q key is on a qwerty keyboard. I learned pinyin but it's abundantly clear that zhuyin is the superior system for foreigners. Ironic no? The bottom line is that pinyin is written in latin letters, where q and x are re purposed to be sounds that don't exist in English. That means that every beginner learner of Chinese in the West ends up using a phonetic way to write Chinese which clashes with their existing notions about how latin orthography ought to be pronounced. It's a big deal, seriously, I've heard third year students continue to pronounce Chinese based on erroneous pronunciation of pinyin. Zhuyin doesn't have that problem because it uses unique symbols, we have no preconceived notion. Oh no, more strange symbols to learn? Well yes, but honestly, the 37 zhuyin symbols are pretty trivial if you're going to learn the thousands of Chinese characters required to attain any level of literacy, right?
The problem is this; Taiwanese Mandarin isn't the same as Mainland Chinese Mandarin. It's not just the character set, you can't just morph one to the other, it's also the lexicon. In the past I used Google's pinyin input system on computers (谷歌拼音輸入法). A godsend because most Chinese authored software drowns in a sea of adverts and potentially nefarious government sanctioned malware. It also freely let you switch between character sets and synchronised your vocabulary across devices via their cloud service. Awesome right? Unfortunately I've found continued use of the Google pinyin system is untenable. Firstly, the software is long in the tooth since Google unceremoniously exited the Chinese market in 2010, so they're not that bothered any more. Updates have been minor and right now, the IME completely crashes Word 2016 when you attempt to use it. No shit. Also the cloud sync just doesn't work and there's no pre-loaded vocab options relating to Taiwan.