To completely give up or abandon a goal, hope, or plan.
When to use
giving up goal, abandoning hope, giving up plan
The nuance
포기하다 is used when one abandons a goal or hope without intention to continue or achieve it.
포기하다 (po-gi-ha-da, pogihada) · 그만두다 (geu-man-du-da, geumanduda)
포기하다 means to completely give up a goal or hope, while 그만두다 means to stop or quit an activity or job.

One lane per word — no nested boxes, just the gist.
To completely give up or abandon a goal, hope, or plan.
When to use
giving up goal, abandoning hope, giving up plan
The nuance
포기하다 is used when one abandons a goal or hope without intention to continue or achieve it.
To stop or quit doing something, such as a job or habit.
When to use
quitting job, stopping habit, halting action
The nuance
그만두다 is used when stopping or quitting an activity, job, or habit without necessarily implying giving up hope or goal.
나는 시험 공부를 포기했다.
na-neun si-heom gong-bu-reul po-gi-haess-da., na neun si heom gong bu reul po gi haess da.
The goal (studying for exam) is completely abandoned, so 포기하다 is correct.
나는 시험 공부를 그만두었다.
na-neun si-heom gong-bu-reul geu-man-du-eoss-da., na neun si heom gong bu reul geu man du eoss da.
Stopping studying temporarily is possible, but 포기하다 sounds more natural for giving up.
그는 담배를 그만두었다.
geu-neun dam-bae-reul geu-man-du-eoss-da., geu neun dam bae reul geu man du eoss da.
그만두다 is appropriate for quitting a habit like smoking.
그는 담배를 포기했다.
geu-neun dam-bae-reul po-gi-haess-da., geu neun dam bae reul po gi haess da.
포기하다 is less natural for habits; 그만두다 is preferred.
회사를 그만두기로 결정했다.
hoe-sa-reul geu-man-du-gi-ro gyeol-jeong-haess-da., hoe sa reul geu man du gi ro gyeol jeong haess da.
그만두다 fits when quitting a job.
회사를 포기했다.
hoe-sa-reul po-gi-haess-da., hoe sa reul po gi haess da.
Jobs are not 'given up'; 그만두다 is correct for quitting a job.
나는 꿈을 포기하지 않을 거야.
na-neun kkum-eul po-gi-ha-ji an-eul geo-ya., na neun kkum eul po gi ha ji an eul geo ya.
포기하다 is suitable for not giving up on a dream (goal).
나는 꿈을 그만두지 않을 거야.
na-neun kkum-eul geu-man-du-ji an-eul geo-ya., na neun kkum eul geu man du ji an eul geo ya.
그만두다 is inappropriate because 꿈 is a goal, not an action.
Which word means 'completely giving up a goal'?
Which word is correct for 'quitting a job'?
Which word fits in ‘quit a bad habit’?
Which phrase is closest in meaning to ‘포기하다 a dream’?