The Framers of the Constitution understood “democracy” and “republic” to mean different things. Explain this difference and identify which concept the Framers favored.

Answer:

Equality" and "justice" mean different things to different people. Equality means what the ancient Greeks meant by the word isonomia -- equal treatment before the law. The U. S. Constitution embodies this view of equality in its "equal protection" and "due process" clauses.

Justice also has two meanings, and the distinction between its two meanings has the same basis as the distinction between the two meanings of equality. First of all, justice is a concept that applies only to human actions, justice refers to actions involving two or more people. An isolated person is neither just nor unjust if his actions do not affect any other people.

The Framers of the U. S. Constitution thought that justice exists when all interactions among people are based on voluntary exchange. What mattered to them was the process of interaction, not its outcomes. This is what the ancient Greeks referred to as "commutative justice."

Frankly they thought democracy was mob rule. They studied history and saw that democracies usually end in tyranny.

James Madison, Federalist Paper No. 10: In a pure democracy, "there is nothing to check the inducement to sacrifice the weaker party or the obnoxious individual."

"Remember, democracy never lasts long. It soon wastes, exhausts, and murders itself. There was never a democracy yet that did not commit suicide." John Adams

A "representative federal " republic is what they concocted instead.
Federalist # 10 explains their rationale.
The Avalon Project : The Federalist Papers No. 10