The real point of resumable exceptions is that the place where the exception happens is often not the right place to decide what to do about it. The OP's missing directory exception may occur inside some low level utility library, which which can be used by many different systems.
If the client code is a batch program then there may be nothing better to do than clean up, log the error and die. However, if the error occurs in an interactive application it makes perfect sense to prompt the user to either create the directory or specify an alternative one and continue. You do not want the utility library to know about the different ways to deal with the missing directory and throwing an ordinary exception loses your current context/state.
With resumable exceptions (or a continuation) you can let client code decide what to do about the missing directory with the option of resuming from where the error occurred, possibly using a different directory.
The utility library just throws a resumable exception (or an exception containing the current continuation) without any coupling to how the issue may possibly be resolved.