They are synonyms. That’s why the Founders called America a “constitutional republic” rather than a “democracy.” “Democracy” is like a color: blue, green, or red. You never find it by itself in the real world. It’s always got something else mixed in: constitutionalism, liberalism, even constitutional monarchy.
And there is a good reason to be precise about using orthogonal terms in orthogonal ways. As Georgetown’s Shadi Hamid explains, conflating “democracy” with “liberalism” and Anglo/European cultural and political norms makes it very hard to understand how democracy actually operates globally: https://www.theatlantic.com/international/archive/2014/05/de....
More to the point here, we don’t have a democracy in the U.S. We have a constitutional republic with myriad checks on majoritarian rule: the Supreme Court, the Senate, the Electoral College, etc. Thus it’s important to be precise about when you’re saying a particular decision should be turned over to the majority and when you think majoritarian impulses must be checked by anti-majoritarian institutions.
Here, I’m trying to be precise: I think the majority should get its way when deciding who to affirmatively support with taxpayer dollars. I think public spending priorities are at the core of where the people are preeminent. That’s why the House of Representatives in the U.S. and the House of Commons in the U.K. have the Power of the Purse. But the majority should be checked by courts when it comes to shutting down independent speech from private individuals.