Who runs the team: the vendor, or you.
A traditional BPO takes a process off your hands and runs it behind a service-level agreement. You buy an outcome; the vendor owns the people, the priorities and the tooling, and an account manager sits between you and whoever is actually doing the work.
Co-sourcing keeps the same offshore cost base but moves the controls back to you. Connext is the legal employer and runs the infrastructure, while you set priorities, meet the team directly and measure quality yourself. So the comparison turns on one question: do you want the process handled, or do you want to run it with cheaper, better-retained people?
Traditional BPO is a sound answer for high-volume, standardized work, and handing a settled process to a vendor is less effort than managing it yourself. Choose Connext when you want to run the operation yourself, at the same offshore cost base, with people who stay.
Follow an engagement from the day you sign it to the day you judge it. At each milestone the two models do something different and at the last one, the BPO wins.

Figures as published on the source comparison page.