Politically speaking, what is the difference between a democracy and a republic?

The two are entirely different and a country can be either without the other.

A republic is any system without a monarch.
A democracy is a system in which the people have a meaningful say in political decisions through votes (direct democracy) and/or elections (representative democracy).

The United Kingdom and Sweden are democracies (all important decisions are taken in a freely elected parliament), but not republics (both have a monarch).
China and North Korea are republics, but not democracies: The head of state is not a monarch and while both technically hold elections, those are not free and have no actual impact on politics.
Saudia Arabia is neither a republic nor a democracy, it is an absolute monarchy.
Iceland and Switzerland are both republics and democracies.

Also, while "republic" is a yes/no decision - there is no intermediate - "democracy" is a gradual term. There are flawed democracies (several European states with right-wing governments that try to restrict the rule of law and freedom of opinion) as well as authoritarian republics with some remaining democratic elements (Russia still has some democratic processes at local level in spite of all relevant authority resting with V. Putin)

https://en.wikipedia.org/wiki/Republic
https://en.wikipedia.org/wiki/Democracy

You may want to just scroll to the bottom!

The definitions of both "republic" and "democracy" are variously understood and have evolved over the past millennium and a half.

Sources speak of "modern republics," but even today we have republics as diverse as the United States of America and the Republic of China. Likewise, modern democracies aren't much like the brief Athenian democracy, but they do always possess the characteristic of being a government "of the people." But compare the Netherlands to Nigeria.

Many democracies, including the US, style themselves as republics.

Look to their etymologies:
"Republic" comes from the Latin "res publica," which means "the common good" or "the people's concern."
"Democracy" comes from the Greek "people-government" or "popular government."

Today, a REPUBLIC is understood to be a government that exists for the welfare of the citizens and is representative in structure. In its various applications, the representatives have sometimes been chosen by the people and sometimes chosen by the leaders. https://www.britannica.com/topic/republic-government

Today, a DEMOCRACY is understood to be a government in which "a nation's citizens have the power to decide the laws under which they will live."
https://worldpopulationreview.com/country-rankings/democracy-countries

Regarding the debate over whether the US is a republic or a democracy, I like how Jay Cost puts it:
"Our system is republican in that the Founders understood that the public is the only legitimate sovereign of government. But it is not wholly democratic, in that they feared the abuse of that authority by the people and designed an instrument of government intended to keep temporary, imprudent, and intemperate outbursts of public opinion from dominating the body politic."
https://www.aei.org/articles/democracy-or-republic/?gclid=CjwKCAiA8NKtBhBtEiwAq5aX2N3bMnHxmhLpxMExg72adg-ybVvLbZSuxd2piYh1It2JdxKME3pSzxoCYiIQAvD_BwE

But perhaps the most straightforward answer for ozzz2002 can be found in the CIA's list of government types:

Democracy - a form of government in which the supreme power is retained by the people, but which is usually exercised indirectly through a system of representation and delegated authority periodically renewed.

Republic - a representative democracy in which the people's elected deputies (representatives), not the people themselves, vote on legislation.
https://www.cia.gov/the-world-factbook/field/government-type/