~(으)면서 and ~다가 ..
I'm confused about those two grammar forms becuz both of them are translated to "while doing something .."
While reading a lesson about ~다가 I found that they are different but I still feel like I don't get it ..
I understood that ~다가 is used when one of the action stops the other one
"동생이랑 계속 얘기하다가 엄마의 말씀을 못 들었어요 " ..
And ~(으)면서 is used to say that someone is doing two action at the same time
" tv를 보면서 콜라를 마셨어요 " ..
Is there any other difference so that it can be clearer ??

Thank you ^-^ ..

Question

~(으)면서 and ~다가 ..
I'm confused about those two grammar forms becuz both of them are translated to "while doing something .." 
 While reading a lesson about ~다가 I found that they are different but I still feel like I don't get it ..
 I understood that ~다가 is used when one of the action stops the other one 
 "동생이랑 계속 얘기하다가 엄마의 말씀을 못 들었어요 " ..
 And ~(으)면서 is used to say that someone is doing two action at the same time
 " tv를 보면서 콜라를 마셨어요 " ..
 Is there any other difference so that it can be clearer ??

Thank you ^-^ ..

Answer

You got that mostly right.

~다가: something unexpected/unintended happens while another action was continuing
Ex:  잠을 오래 자다가 버스럴 놓쳐버렸다. (I missed the bus having overslept myself)

~면서: one action goes on while another continues in parallel
Ex:  공부하면서 음악을 들었어요. (I listened to music while studying)

* Another difference is that ~면서  has an additional usage of contrasting two states/facts.
Ex: 그는 부자이면서(도) 인색하다.  (He is miserly even though he is rich) 
Ex: 슬프면서도 달콤했다. (I felt sweet while sad at the same time)

This usage has the sense of "even though", "while", or "at the same time".
In this case, it has more to do with a state or condition than a continuing action.