Section 8 housing is a voucher program which provides housing subsidy in the private rental market. The voucher holder finds an apartment on their own in the private rental market and the owner is provided a rental subsidy based on the voucher holders household income. The voucher holder then pays the remaining balance in rent from what the SSHA subsidy payment was. Vouchers are portable meaning that if you move away you can take the subsidy with you.
Public housing assistance means that you receive housing subsidy in SSHA owned apartments only (Stonequist, Jefferson and Vanderbilt Terraces). The amount of rent you pay is also based on your household income (approximately 30%). When you move from public housing, the rental subsidy does not go with you.