Writing is important for communication. Without written communication, many of our businesses and institutions would thrive because it is a non-intrusive form of communication (unlike audio, which we're bound to listen to even if we don't want to; imagine all the billboards on the road screaming their messages at you).
Good writers also make for much comprehensive reading material and help in the lingual development of our society's children.If writing is wiped out of our society, we won't have a strong base for language and will be poor at communicating to one another.
Also, for record-keeping and documentation legal writers are very important who need mastery of legal language and the layman language itself and hence for it to be written down; it is hard to record or document all information as voice (a divorce paper swapped with a divorce recording is not as authentic and is a waste of time and money).
With diminished trend of writing handwriting will also become less legible and of course, communication will be weaker.
And there are a lot more but I hope these at least gave you a good head start. =D
