Public Housing is owned and operated by the government. It is usually apartment complexes. Public Housing is for eligible low-income, elderly, and disabled persons. Typically, you apply for housing through your local Housing Agency. If you are accepted, the agency will either place you on the waiting list or place you in housing if it is currently available. If you are denied, you can appeal and ask for a hearing to discuss the denial.
Tenant based Section 8 gives you a "voucher" which can be used to rent any kind of housing (house, apartment or condominium) and with any landlord that is willing to accept it. If you receive a voucher, you then have to go look for a home yourself. There are time limits for finding a home and the voucher can expire if you do not find a home in time. If you find someplace you want to rent and the landlord is willing to accept your Section 8 voucher, you then notify Section 8. If the home meets housing quality standards, reasonable rent standards, and other program requirements, then Section 8 should accept the home you found. If you decide to move out of this type of Section 8, your voucher goes with you and you can look for a new home to rent.