I think they both mean that you have to do something because there is a social pression like なければならい、but I'm not certain that I can use them interchangeably. Could somebody confirm it?
Come one natives! My best guess (yes, please downvote, as this site is not for guesses, but I hope @Alex16 and I at least get someone to answer ;-) )
ないわけにはいかない has more of of the social pressure aspect (you are supposed to do X) while ずにすまない more likely means that there are some more "rational" (*) reasons for why failing to do X would cause problems.
(*) While it is "rational" to consider the other people when adjusting your message, here, by "rational", I meant simply a statement made without considering the "public opinion"