People say this all the time to try to sound wise, but in reality it's sort of a meaningless distinction.
There's a wide range of functioning models for democracy, and a democratic republic is just one form of representative democracy:
Nearly all modern Western-style democracies function as some type of representative democracy; for example, the United Kingdom (a unitary parliamentary constitutional monarchy), India (a federal parliamentary republic), France (a unitary semi-presidential republic), and the United States (a federal presidential republic).
