마음(맘) means usually emotion or mind. For example, "그 남자(That guy) 마음(mind) 속에 무엇(what)이 들어있는지 도대체 읽을 수가 없어.(=It's totally impossible for me to read that guy's mind.)" 맘 is contraction of 마음. Though 맘 and 마음 are interchangeable, 마음 is more preferred than 맘 in writing. Literal meaning of 심장 is heart, the organ placed in left chest and pump the blood by regular beating. However, when 심장 is used in a lyrics, a poem, or an emotion, it has figurative meaning. For example "어제(yesterday) 그(him)와 헤어지고 나니 심장(heart)이 너무아파. (=My heart hurts so much(I'm feeling blue so much) after breaking up with him(그) yesterday.)" In a lyrics such as "나는 심장이 없어 = I don't have heart(I can feel nothing now because of too much sorrow)" In short, 심장, 마음(맘) are different in literal meaning, whereas 심장 and 마음(맘) can mean the same thing in figurative expression.

Heart is translated in Korean as "마음" and "심장" depending on its references. When we refer to heart as the organ in our chest, we mean "심장", and when we refer to heart as the place (I can't come up with a better word) for emotions, we mean "마음." "My heart is beating so fast. " In this sentence, my heart means "내 심장" or "내 가슴", which means "chest." "You are breaking my heart." In this sentence, my heart means "내 마음." I hope this helps.