Why do people keep saying this? The two aren't mutually exclusive.
We keep saying it because they're radically different from one another.
A Democracy directly implements the will of the majority. 51 out of 100 want something? They get it. Simple. Appealing until you realize that often, the 51 want slavery, religion in government, control over your sex life, retarded limits on contracts such as marriage, to deny health care to you because you're not wealthy, etc. Best not to go there. The old saw "Democracy is two wolves and a lamb deciding what's for dinner" about covers it.
Whereas a republic doesn't implement the will of the majority at all; it implements the will of the representatives. The idea being that the representatives are honorable, thoughtful people guided by principles more sophisticated than the masses, if for no other reason than because they actually have the time to think things through, as this is their day job, as it were. The concept of a working republic depends utterly on the consistent selection of a majority of honorable, thoughtful representatives who are able to gather sufficient truths about the matters they must create legal structures for (if you're thinking "uh-oh..." then you get a gold star.)
In the specific case of the USA, which is a constitutional republic, by design it implements the will of the representatives, moderated by the constitution's fundamental limits and enumerated powers. Furthermore, the US constitution insists (in article 4, section 4) that each state government is also implemented as a republic. The limits and enumerated powers of the constitution were intended to prevent abuses such as those I outlined as typical for a democracy.
Design aside, the actual function of the US federal government is implementation of the will of the representatives as dictated to them by moneyed and powerful special interests, very rarely moderated by the constitution, then further (as a matter of fiat power grab) moderated according to the whim of SCOTUS, which, thus far, has often been quite at odds with the plain and obvious requirements of the constitution. Technically, this actually turns out to be an oligarchy, something quite prone to abuses, as we see demonstrated in a most concrete manner on a daily basis.
The difference between a democracy and a constitutional republic is immense enough. But the difference between a democracy and the oligarchy we actually have is almost incomprehensible.
What is most likely confusing you about our (nominal) republic is that small portions of the process appear to be somewhat democratic in nature. For instance, once the power brokers in the political parties pre-select the special-interest-compliant figureheads we get to vote for, we can, quite democratically, select either one from column A or one from column B. Most other portions of the process are not democratic. For instance, the FCC, the FDA, the DEA, the CIA, the FBI, the reserve banking system, SCOTUS... these are not democratic institutions, they exist in forms almost completely insulated from the democratic process. For instance, many functionaries persist across voting cycles for representatives; some, like SCOTUS, are effectively impossible to dislodge; some entities, like the federal reserve system, exist outside effective control of anyone at all.
The bottom line, though, is perfectly clear: The USA is not a democracy, and has never been one.
If nothing else, take your hint from the pledge to the flag: "...and to the republic, for which it stands..."