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If the question is "What's the difference?", then, as is tradition, we must figuratively clear our throats before such discourse with the well-worn adage, "It depends."

As a disclaimer, much of this content was copied from Wikipedia and arranged in a way to support my opinion; none of this should be taken as Gospel. This is not financial advice. And please accept my apologies for the tedious length.

If we limit our terms' definitions to their etymological roots, then:

Democracy

• The term democracy first appeared in ancient Greek political and philosophical thought in the city-state of Athens during classical antiquity. The word comes from dêmos '(common) people' and krátos 'force/might'.

• In a direct democracy, the people have the direct authority to deliberate and decide legislation. In a representative democracy, the people choose governing officials through elections to do so. The definition of "the people" and the ways authority is shared among them or delegated by them have changed over time and at varying rates in different countries.

Republic

• The term originates from the Latin translation of Greek word politeia. Cicero, among other Latin writers, translated politeia into Latin as res publica, and it was in turn translated by Renaissance scholars as republic (or similar terms in various European languages). The term can literally be translated as 'public matter'. It was used by Roman writers to refer to the state and government, even during the period of the Roman Empire. The term politeia can be translated as form of government, polity, or regime, and it does not necessarily imply any specific type of regime as the modern word republic sometimes does.

• A republic, based on the Latin phrase res publica ('public affair' or 'people's affair'), is a state in which political power rests with the public (people) through their representatives—in contrast to a monarchy. Although a republic is most often a single sovereign state, subnational state entities that have governments that are republican in nature may be referred to as republics.

• Representation in a republic may or may not be freely elected by the general citizenry. In many historical republics, representation has been based on personal status and the role of elections has been limited. This remains true today; among the 159 states that use republic in their official names as of 2017, and other states formally constituted as republics, are states that narrowly constrain both the right of representation and the process of election.

• The term developed its modern meaning in reference to the constitution of the ancient Roman Republic, lasting from the overthrow of the kings in 509 BC to the establishment of the Empire in 27 BC. This constitution was characterized by a Senate composed of wealthy aristocrats wielding significant influence; several popular assemblies of all free citizens, possessing the power to elect magistrates from the populace and pass laws; and a series of magistracies with varying types of civil and political authority.

Plutocracy

• A plutocracy (from Ancient Greek πλοῦτος (ploûtos) 'wealth' and κράτος (krátos) 'power') or plutarchy is a society that is ruled or controlled by people of great wealth or income. The first known use of the term in English dates from 1631. Unlike most political systems, plutocracy is not rooted in any established political philosophy.

• Some modern historians, politicians, and economists argue that the U.S. was effectively plutocratic for at least part of the Gilded Age and Progressive Era periods between the end of the Civil War until the beginning of the Great Depression. 

• President Theodore Roosevelt became known as the "trust-buster" for his aggressive use of antitrust law, through which he managed to break up such major combinations as the largest railroad and Standard Oil, the largest oil company. According to historian David Burton, "When it came to domestic political concerns, TR's bête noire was the plutocracy." In his autobiographical account of taking on monopolistic corporations as president, Roosevelt recounted:

...we had come to the stage where for our people what was needed was a real democracy; and of all forms of tyranny the least attractive and the most vulgar is the tyranny of mere wealth, the tyranny of a plutocracy.

On paper, we (the U.S.) are a not a direct democracy, though we do vote directly about some issues via referendums; our constitution codifies the extents and limitations of legislation, enforcement, and jurisprudence of our laws and our rights as citizens.

We directly elect representatives to carry out the business of governance from local, state, to the federal level as our country's political union is a federation of States that simultaneously retain their autonomy via the parameters outlined within the constitution and cede ultimate authority of jurisprudence to our bicameral national assembly (in our case, Congress) and Supreme Court.

In practice, due to regulatory capture, political expedience and corruption, and the realities of our global economic expansion, our country is effectively ruled by 2 factions of a political class of wealth that use faux-populism to maintain their power and influence.