This is a question which comes up periodically in some of the heraldry-oriented societies, newsgroups, and other organizations to which I belong. One regularly-recurring answer, of course, can be described as "snobbery". The idea is that people want coats of arms to show that they are better than the other folks around them, rather like being a member of an exclusive club. Not just anybody can get in.
While this is an answer that may apply to some folks who want to use heraldry, I think that for most people, it’s a little too simplistic. That is, I think many people’s interest in heraldry is far more complex, and interesting, than just a desire to set themselves apart from the riff-raff.
A couple of quotes that I’ve run across help to demonstrate this, I think:
"Why do people in the twentieth century still want an outmoded means of personal identification? The answer is, I believe, ... people are still looking for ways of laying down a solid foundation for their family. Heraldry is traditional; it is also fun, colourful, historic and respected." (Peter Spurrier, The Heraldic Art Source Book)
"I was mildly familiar with heraldry (I don’t think I even knew what the name was when I first started actively attempting to understand it...) and it dawned on me one day that these ‘family crests’ are powerful familial emblems that I could appropriate in establishing my own family traditions. ... I’ve found that friends are very interested in my coat of arms. I usually don’t say anything about it but when they see my framed COA [coat of arms] in my house or see my desktop background at work they start asking a lot of questions. After I explain that I’m not royalty, I haven’t been knighted and so on, many of them say, ‘That’s exactly what I want to do. It’s called heraldry? I’m going to check into that.’" (Sanjay Merchant, American Heraldry Society Forum, January 7, 2005)