If you haven't read chapter 27, I'd suggest you go read it before you look at what I have to say.
It's most probably due to Sangwoo seeing Bum as his "mother" or bearing some resemblance to her (but we don't know that for sure since we don't exactly know how his mother actually looks like). In chapter 27, there are bruises seen on the feet of the mother as she was calling for Sangwoo, which indicates that she was abused by Sangwoo's father (but we already know that after that chapter).
This is just what I think, but Sangwoo may be depending on Bum and doesn't want him to leave his side just like his mother did. I'm assuming that Sangwoo's mother was either abused to death, which is why he is so afraid of Bum dying on him, or that his mother had committed suicide being unable to bear the abuse any longer.
But then again, I could be wrong or am mostly wrong.
Does Sangwoo have a case of bipolar disorder or something? I was literally in cold sweat when I was reading chap. 27 from the links sent here.