A Republic is a state that is not ruled by a monarch, but by group of people who are citizens, voted in via some process. Article IV, section 4 of the US Constitution guarantees to every State of the Union a Republican form of government. This means, I think, 1) there will be no monarch, and 2) the leaders are themselves ordinary citizens (and therefore subject to the same laws as every other citizen), and 3) that the leaders are selected by the people.

A Democracy is a government formed by election of the citizens. In ancient greece, not everyone was a citizen. Slaves were not citizens. I do not believe that women were full citizens. While the ancient greeks pioneered democracy, and their government is recognizable as democratic today, it was not democratic in the way that our government is democratic.

1. As we did originally, so they had slaves.
2. Their democracy was direct, while ours is representative. With the exceptions of referenda in localities, we don't vote on issues. We elect representatives who vote on the issues. We don't vote for a president. We vote for seats in the electoral college.
3. The ancient Greeks had a powerful mechanism for protecting their democracy from the undue influence of entrenched power. (Look up the etymology of the word "ostricize," for example.)

Democracy and Republic are neither mutually exclusive, nor entirely synonymous. They are complementary.