They are close aren't they?
벨 소리가 (나와서/나서) 문을 열었어요.
벨 소리가 나와서 문을 열었어요. The bell sounded (out) so I/he/they opened the door. Alt: The door was opened at the sounding of the bell. ~when/because the bell rang (out).
벨 소리가 나서 문을 열었어요. The bell rang *(sounded)
so I opened the door. Alt: Because the bell rang (he) opened the door. -or- The door was opened when/because the bell rang. *You can still use 'sounded' but the nuance of 'emanation' is absent from the second sentence. BTW - JFF (Just For Fun) In these examples
서 and
so can serve similar purpose - isn't that great!
나오다 -- come out, come forth from, emanate, flow out from
*나다 - (according to your sample sentence) sound, ring
*나다 *also has many other definitions such as: 'happen' 'become' 'appear' 'improve (sickness/situation)' and sometimes 'get' as in 'I
got a pimple.'
That's all I can think of now -- waiting for Dr. Sean !woot!