There are web server packages on node that do catch errors that happen in synchronous code. For async, you can use promises, which take errors into consideration.
I'm not sure what happens on unhandled async errors on other languages, I guess in Java you could have a dangling request (happened to me before) or it could crash the app (also happened to me before). C++? I haven't really done much webserver work on that one so someone else may be able to write more about it.
Also obviously PHP, you just get a white page or actual full error stacks sent to the client, the joy! (edit: of course, if you don't handle it properly)