A payroll job is one where the worker is an employee of the company for which they work. Taxes are withheld and paid into the government. The company pays taxes as well on the worker both as contributions to Medicare, social security and unemployment taxes to paid the worker if they are fired without cause. The worker is also eligible for company benefits if offered by the company. In some States or localities, there may be mandated benefits available in addition to the Federally mandated benefits for employees including overtime and unemployment benefits.
Off-payroll jobs mean you are not an employee. In all likelihood, the worker is an independent contractor. The worker is responsible for all of his taxes. No company benefits and no mandated benefits or protections that are available to employees. However, there are some tax advantages available to an independent contractor such as deducting commuting expenses and all business related expenses. Those deductions are not available to an employee.