I think the better words than "important" to describe what I utterly desire are "significant" and "irreplaceable", and I wish to be significant and irreplaceable to a few people in my life, my parents, my partner, my children, my best friends, even my close workers.
I loathe being described, thought of, talked about, talked to, treated, loved, cared for like everybody else, or anybody else. For myself and within every person I've connected to, I must have an authentic identity.
Hence it's easy for me to feel unimportant or insignificant. Most of the time, people don't put much thought and energy into truly identifying the people in their lives. Not everyone allows themselves to dive truly deep into another person to discover their uniqueness; people settle with their own reflection on somebody else. And so after a while a person you know is just another face. That IS scary.
When I leave a work place, if I hear that I'm hard to replace for the detailed good work that I've done, I know I'm important. When I'm with a partner, if I'm aware that I make him feel and do things he has never before, and the things we do / say are different and beyond what's between him and his previous, I know I'm important. When I go home after a night out with my friend and they let me know that it's been a long time since they could talk / have fun with somebody that way, I know I'm important.
Say, it's an INFP type 4 thing?!