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.
