They existed primarily for monospaced font display in old computing contexts. But now half-width kana generally shows up only as an aesthetic thing or in very specific settings where space must be saved.
Full-width Latin characters show up plenty in typed Japanese text, it doesn’t mess up the monospaced alignment and I suppose the Latin text stands out a bit more. But you’ll also see it written ‘normally’ plenty.
As for typing it yourself, maybe on some site registration forms…? I can’t remember ever having to use half-width, outside of Japan… Just use full-width hiragana and katakana like normal. (タンタン is katakana btw)
I often have to use full-with romaji when I enter my name in Japanese forms that require separate fields for name and furigana. Generally Japanese people just use kanji, hiragana, or katakana in the name field, and those are all full width. They don’t expect half width romaji I guess.