"Whether you call it a republic or democracy in the English language the terms are interchangeable."
Exactly! A republic is properly the opposite of a monarchy, not a democracy. A republic typically uses one form or another of democracy; we, of course, use a representative democracy (except for ballot initiatives which are a form of direct democracy). People who make this statement typically assume that democracy only means direct democracy, where the voters decide all rules and have a "mob rule" approach, but there are other forms including ours. Correctly stated, we are a constitutional republic that is a representative democracy.
This whole "we are a republic not a democracy" statement is a cutesy meme which just demonstrates that the speaker doesn't really know what he's talking about but heard this in right wing / libertarian circles and thinks it makes him look smart.