Q1. "Why are there two standards?"
According to Wikipedia (https://simple.wikipedia.org/wiki/Zhuyin) , Zhuyin was the first phonetic system in use from 1911 after the xinhai revolution.
After the overthrow of China's last emperor during the Xinhai Revolution in 1911, the new government in China created Zhuyin to help the common people read more easily.
It was the communists who wanted romanization and to do away with the Chinese writing system entirely. (https://www.newyorker.com/magazine/2004/02/16/oracle-bones)
However, the Chinese Communist Party, including Mao Zedong himself, wanted to ban writing Chinese characters altogether and replace them with the Latin alphabet.
Q2: "Does each have a specific context/setting in which they are used?"
When the People's Liberation Army defeated the Kuomintang (the founding party of the Republic of China) in 1949 and sent them off to exile in Taiwan, the use of Zhuyin dropped in mainland China because the CPC was interested in using the Latin alphabet for writing Chinese phonetically. However, Zhuyin is still widely used in Taiwan as it is used to type Chinese on computer and phone keyboards.
So today, Zhuyin is mainly used in Taiwan and Pinyin is mainly used in China and by many foreigners studying Chinese, though it seems (personal observation alone) a sizable number study zhuyin, too.
Q3: "Which one is more popular in general?"
Popular? I'm not sure how to gauge that.
But in terms of sheer number of users, mainland China's pinyin clearly has the greater number of users just through population alone.
- I feel zhuyin more closely matches Mandarin phonetics.
- I prefer reading ㄘㄢ to "can" which I want to pronounce like English "can"
- And I've used zhuyin to reinforce tones when typing.