Both of them are designed to help businesses with their software development needs.
In the staff augmentation model, the company manages the external resources. They will be just adding some additional resources to the team while being at the top of management.
In the case of the outsourcing model, the company gives away the responsibility of managing the tasks to the outsourced partner. They then take a backseat or invest their time and effort in core business areas, while the outsource partner will be managing IT needs on their behalf.