脑手术 and 腦手術 are both Chinese (simplified and traditional), the latter of which looks the same in this font as the Korean (because the third and fourth strokes on the left-most radical have become horizontal rather than diagonally down and up respectively) except for the space which I suppose is what makes it Korean? 脳手術 is the Japanese which is also a bit different yet . . .
뇌 수술: brain operation. It is the surgical operation.
I have a brain surgery tomorrow = 내일 뇌 수술이 있어요.
뇌외과: medical field in brain surgery (Not 100% sure but I think it is more medically accurate to say 신경외과, which is "Neurosurgery", while 뇌외과 is used colloquially).
This hospital has the best brain surgeons = 이 뱡원에는 제일 좋은 뇌외과 의사가 있어요.
*The Korean sentence examples I've given may not sound natural; I only used what little I've learned.