I'm in NYC and this is what we refer to as the projects here: https://i.imgur.com/UyTKWJg.jpg It's mostly people living on welfare but there's all sorts of people living in these. People that are disabled, poor, people with money but want to be cheap and save money. The buildings and the elevators can sometimes smell like piss. On garbage days, the streets and the walkways in between are covered in litter.
There is overlap, in project-based Section 8, which is based on government contracts with private landlords for specific housing projects, as opposed to the better known form of Section 8 which follows the individual rather than being locked to a particular housing project.
My understanding is that these are logistically different things. Public housing projects are state owned, whereas Section 8 refers to privately owned housing where rent is paid at least partly by government vouchers.
It's one of those technically different, but functionally the same things.
"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.