Both types of activiites aim at providing employees to business.
The difference is the period for which such employees are provided.
Staffing is about providing temporary employees to businesses in need of additional help for a short period of time, typically offices, warehouses, factories, retail stores or other customer service positions. The length of time a temporary employee remains depends on business need.
Placement services specialize in filling jobs within specific industries in permanent positions. This is a broad term which includes employment agencies, in-house or corporate recruiters, headhunters, and consulting firms.