신경 means "nerve" 걱정 means "worry, concern, care, anxiety" 신경(을) 쓰다 means "bother; show concern, mind" 걱정(을) 하다 means "worry (about), be concerned[worried, anxious] (about), feel[be] uneasy (about)"

신경 is concern. While, 걱정 is worries. 신경 써요 = to put concern/care/notice about something 걱정해요 = to put worries about something