Today, only 割 is the really productive word for "one tenth". 分 is not the first choice.
Mostly as the linked answer said, 分【ぶ】, in the modern language, is either:
- supplementary unit that follows various main units (including 1% in ~割~分), or
- fixed term in certain measurements (see dictionaries), or
- part of set phrases (五分五分, 九分九厘 etc)
If I have to name a rather productive usage, 分 is sometimes used independently to mean 10% in the context of relative length or degree of completion, because 割 tends to remind people of "pie chart" while 分 of "progress bar"* (but no problem replacing with 割 either).
Aside from that, there is another confusing aspect regarding 分 that it also has a minor sense "one n-th" in some technical terms:
- (typography) 二分アキ (1/2 em space), 三分アキ (1/3 em space), 二分[四分]{しぶ}アキ (3/4 em space)...
- (music) 二分音符 (minim), 四分{しぶ}音符 (crotchet), 十六分音符 (semiquaver)...
*: This is why I think 五分五分 comes to mean "fifty-fifty" and not "half-and-half".