In my 40 years I've never seen a restaurant that called itself Iranian, only Persian.
Of course, in my lifetime, politics have been such that the distinction makes sense in terms of restaurateurs not wanting to scare Americans (who may or may not be quite aware of the CIA's role in shaping the Iranian government of today) off. But if I understand correctly, it may also be a distinction that they themselves care to make precisely as a personal political statement. Or for that matter a culinary one, the cuisine having ancient roots.
Can anyone speak to this?