No there is a huge difference between a federal republic and a democracy in my opinion. In the original design most of the power of government was concentrated in the states. That allowed the people in those individual states ready access to their government. The states gave up very few powers to the national government, in fact Art. 1 Section 8 of the Constitution of the USA outlines them specifically. All other powers were reserved to the states themselves. Since the states created the national government and reserved all other powers to themselves they were originally designed to be the more powerful entity.
What has happened, particularly since the beginning of the 20th century is that the central authority has seized more and more power leading to the imbalance we now see. Additionally, the balance of power has shifted from the legislative branch to the executive. Further, the representation to person ratio has gone from 1:20,000 at the time of ratification to 1:750,000. That means that we are no longer a representative republic, a democracy, nor do we have any semblance of even democracy. We are an oligarchy at best and a tyranny at worst.