How do "Bopomofo" and "Hanzi" differ in terms of writing Chinese?

漢字/汉字 (hànzì, trad./simpl.) means Chinese characters. Japanese Kanji and Korean Hanja are also 漢字, meaning Sino-Japanese and Sino-Korean characters in this case.

Bopomofo aka Zhuyin Fuhao is a semi-syllabery transcription of Mandarin Chinese based on ancient oracle bone inscripts as reading aid for kids and foreigners and for certain characters which have two or more readings. In China it was replaced by the romanization standard Hanyu Pinyin in the late 1950s, in Taiwan it is still used but also replaced by Hanyu Pinyin or a kind of Taiwan Pinyin more and more, because Bopomofo requires a Chinese keyboard like this:

One example for characters where Bopomofo, written vertically next to it, may be needed or useful, is 碁 (jī) in the Chinese name of the Taiwanese computer vendor Acer, which is 宏碁 (Hóngjī, “splendid base”) in traditional Chinese.

碁 is normally read as qí, meaning chess or chess-like game such as the Japanese Go, where you may have seen the character.

In simplified Chinese there is no 碁, the character for chess is written 棋 (qí) and 碁 in the sense of basis or fundament is 基 (jī). Therefore Acer’s Chinese name in Mainland China has been changed to 宏基 (Hóngjī).