The difference is important, as Bradley J. Birzer explains over at The Imaginative Conservative: Americana Res Publica: No Revolution. As Birzer writes:
[T]he Republic was neither purely a commercial nor libertarian one. Indeed, the American founders crafted not a commercial republic, but a virtuous republic, allowing for commerce and liberty to serve as a means by which man could use each of his gifts wisely and for the common good (the good thing; the res publica).
While not all of the founders belonged to orthodox Christian denominations or even subscribed to Jewish or Christian orthodoxy, they each accepted most of what the Judeo-Christian context and heritage had bequeathed to them.
Their understanding of liberty was not the collectivist or primivist liberty of Rousseau or the atheistic and abstract liberty of Locke, but the liberty of St. Paul as described in his letter to the Galatian Christian community, the freedom to do what one ought to do.
For most patriots, one could find the best definition of liberty in the prophetic writings of Micah (4:4), as our own John Willson has reminded us many times. “But they shall sit every man under his vine and fig tree, and none shall make them afraid,” the Jewish prophet had written.The distinction between a democracy and a republic has been lost on most of us in the modern period. It would be salutary if we could recover the idea of republicanism (with a small "r") that motivated most of our Founders.