Actually it depends. When Neil Bush's company Ignite! opens a branch office in Mexico to do work previously performed in TX, that would offshoring (sorta). If his company contracts with a Mexican firm to do the work, that's outsourcing.
IBM outsourced the development of an operating system for their PC, hence the birth of the evil empire and rise of the world's richest Harvard dropout. Today IBM is rapidly developing branch offices in China, India, and Pakistan as replacements for US employees, that's offshoring.
Halliburton (particularly KBR) is being paid to perform many duties previously performed by uniform personnel that's outsourcing. But unlike private firms, DOD isn't saving us much money through this practice.