When I was in public schools a few years ago, I learned the difference. Its just that in much of "public" language, you get people interchanging the two. Why is this? Probably has to do with the fact that many people simply forget alot of their schooling once they leave.
I minored in physics in college. I told this to someone and they asked me what physics was. I was taken aback and she said it had been a long time since she was in school. After I simply said that it was study of the physical world(as a short to the point definition that really needs more expanding on but is understandable to anyone), she said she felt dumb.
As to the topic at hand. The simplest way to think of the difference between the two is that:
Democracy: the people vote on everything directly, from laws to anything else
Republic: the people vote for representatives that make the laws and anythings else
Obviously, you can go more indepth than this, but this would be the simplest and easiest to understand for anyone.
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