"The projects" on the other hand are owned and operated by government.

"Projects" and "Section 8" are two separate government programs, but they both wind up with similar demographics because it only attempts to solve one issue.

And in the case of Section 8, assumes best faith action of a third party. Which is hardly the case. Often, an apartment complex gets built as "Section 8" housing. And since the government is footing the bill, not the tenant, it's built to the least possible standard and service is usually poor. Because the incentive for the landlord is to spend as little as possible on it.

Is it synonymous with a Ghetto?

"The projects" strictly refers to the building developments themselves, but is often used informally to refer to a poor neighborhood in general.

"The Ghetto" refers to a neighborhood of an urban area and not a particular building or group of buildings. "The projects" are often in the ghetto but not necessarily.

Saying "I grew up in the ghetto" and "I grew up in the projects" are more-or-less equivalent. Both only refer to urban areas; there is no such thing as a rural ghetto.

The term rural ghetto can be used.

https://en.wikipedia.org/wiki/Rural_ghetto

I didn't say that rural poverty isn't poverty. I'm simply saying that the word "ghetto" almost always refers to an urban context.

”the word "ghetto" almost always refers to an urban context.”

These are not the same thing.

Project refers either to public housing, or more generally any group of buildings where economically disadvantaged people tend to be segregated.

Because these very frequently overlap in the US, people often use them interchangeably.