The U.S has more or less functioned as a Representative Democracy with a very loose Republican Framework in practice since the early 1800s after the States started expanding the Franchise. In part because since the people guaranteed the right to own firearms but who couldn’t vote under the system as originally envisioned. Also decided they wanted the right to vote.

As originally envisioned. Most of the population including most Whites could not vote. And the States could and frequently did arbitrarily raise or lower additional barriers to voting of their own accord.

So technically on paper at least. There’s a difference. In reality we’ve functioned more or less as a representative democracy for most of our history. Instead of the Aristocratic Republic originally envisioned. Which was more or less a dead concept with the first expansion of the right to vote.

Ironically the steady expansion of the Franchise was a point of contention and concern for some of the founding fathers in their later lives after they retired from public life. As Democratic influence on the Republican experiment rapidly spiraled far outside the scope of what they initially intended.