The act of taking something out from inside
When to use
Taking items from bag, Bringing up a topic, Taking papers from drawer
The nuance
꺼내다 is used when naturally or simply taking out objects or words; it does not imply much force or complexity.
꺼내다 (kkeo-nae-da, kkeonaeda) · 빼내다 (ppae-nae-da, ppaenaeda)
꺼내다 generally means to take something out from inside, while 빼내다 implies actively and forcefully extracting something out.

One lane per word — no nested boxes, just the gist.
The act of taking something out from inside
When to use
Taking items from bag, Bringing up a topic, Taking papers from drawer
The nuance
꺼내다 is used when naturally or simply taking out objects or words; it does not imply much force or complexity.
The act of forcefully or cleverly extracting something out
When to use
Removing parts from machine, Secretly extracting documents, Poaching employees from competitors
The nuance
빼내다 is used when physically exerting force or using strategy to extract something, including sneaking something out.
가방에서 책을 꺼냈다.
ga-bang-e-seo chaek-eul kkeo-naess-da., ga bang e seo chaek eul kkeo naess da.
Taking a book out naturally from a bag fits 꺼내다.
기계에서 부품을 꺼냈다.
gi-gye-e-seo bu-pum-eul kkeo-naess-da., gi gye e seo bu pum eul kkeo naess da.
Removing parts usually requires force, so 빼내다 is proper here.
회의 중에 중요한 이야기를 꺼냈다.
hoe-ui jung-e jung-yo-han i-ya-gi-reul kkeo-naess-da., hoe ui jung e jung yo han i ya gi reul kkeo naess da.
Bringing up a topic naturally in conversation suits 꺼내다.
비밀 문서를 몰래 꺼냈다.
bi-mil mun-seo-reul mol-rae kkeo-naess-da., bi mil mun seo reul mol rae kkeo naess da.
Secretly extracting implies 빼내다, not 꺼내다.
서랍에서 서류를 꺼냈다.
seo-rap-e-seo seo-ryu-reul kkeo-naess-da., seo rap e seo seo ryu reul kkeo naess da.
Taking papers naturally from a drawer fits 꺼내다.
경쟁사에서 직원을 꺼냈다.
gyeong-jaeng-sa-e-seo jik-won-eul kkeo-naess-da., gyeong jaeng sa e seo jik won eul kkeo naess da.
Poaching employees requires strategy or force, so 빼내다 is correct.
문제의 핵심을 꺼내서 설명했다.
mun-je-ui haek-sim-eul kkeo-nae-seo seol-myeong-haess-da., mun je ui haek sim eul kkeo nae seo seol myeong haess da.
Explaining by naturally bringing up the core fits 꺼내다.
기계에서 부품을 힘들게 빼냈다.
gi-gye-e-seo bu-pum-eul him-deul-ge ppae-naess-da., gi gye e seo bu pum eul him deul ge ppae naess da.
Forcefully removing parts fits 빼내다.
Take out a book from the bag?
Secretly extract documents?
Take papers from drawer?
Poach employees from competitors?