For the same reason people think being a Republican or Democrat matters?
Nifty50 says: "If a voter does not have a basic understanding of what kind of government they live under or how it works, do they really have any business voting? My answer is absolutely not!"
is less than half correct. Having a basic understanding of how our government works doesn't mean labeling it, and it doesn't mean knowing the difference between republic or democracy. It means what it says... no more, no less.
Point of fact: We're not living in a Republic by the strictest definition. The best you can say is that it's got more elements of a Republic than it has of a Democracy, but that doesn't really tell you anything.
As a people, we're not going to get anywhere until we stop getting hung up on what to call something and get down to business on what we want as a people and what we know works and doesn't work. In my experience, people who throw around terms like republic or democracy to identify our government are looking to say that one of them is bad or one of them is good. It's the same kind of mentality that wants to label Obama as a socialist, because if you can successfully do that, then socialism = bad and therefore Obama = bad. It's much easier than dissecting his policies and PROVING that he's a bad president. (and in case you think I'm plugging Obama, I think he is an idiot. I didn't vote for him the first time and I don't plan to vote for him next time either)
It's academic to talk about our country as a republic or democracy. If you're making an academic point, then by all means, go for it. However, if you're trying to make a political point, even assuming you prove your point one way or another, what does that really mean? You've still got to prove why one is preferable over the other, and it's not as easy as you might think. It's much more constructive to root out what is bad about our system and how to improve it. It's also much simpler to ignore the label and go for the substance in the first place.