감격스럽다 means to be impressed or touched deeply.
뭉클하다 means to be touched and moved deeply.

This sounds the same to me. What are the differences between this 2?

Is 뭉클하다 always used with 가슴이 ?

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Question

감격스럽다 means to be impressed or touched deeply.
뭉클하다 means to be touched and moved deeply.

This sounds the same to me. What are the differences between this 2?

Is 뭉클하다 always used with 가슴이 ?

도와주세요 
고맙습니다

Answer

감격스럽다 can be used in the situation you feel very happily overwhelmed.

뭉클하다 is for the mood when you feel sad in a way. And it is your feeling. So you can’t say the movie is 뭉클. You should say 영화를 봐서 마음이 뭉클하다.(가슴이 뭉클하다is also fine)

영화가 감격스럽다 = 영화를 봐서 뭉클하다  
선물이 감격스럽다= 선물을 받아서 뭉클하다

They are not exactly same, but can be used in similar way. Hope it will help you.