I think anyone who becomes a screenwriter inevitably makes his peace with that anonymity, the same way that most of us make peace with being asked, as we often are, by casual friends and family members -- "So, when are we going to see your next movie?" -- something which, for the overwhelming majority of screenwriters, only happens at very long intervals -- if you're lucky every four or five years.
On the other hand, once you get outside the world of major film fans, I doubt that most people even know the names of any directors, beyond Stephen Spielberg and George Lucas and maybe one or two others, tops.
I doubt most people, if you asked them what a screenwriter was, or did, or what a director was, or did, or what a producer was, or did -- that they would even know, even those who watch the Oscars year after year and see the awards handed out.
NMS