Buying capacity versus buying an outcome.
These are two different purchases, not two vendors competing for the same job. Staff augmentation adds people to your team: they work inside your systems, on your priorities, and you own the outcome. BPO hands a whole function to a provider who manages it end to end and owns the KPIs.
The comparison turns on where you want responsibility to sit. Neither model is Connext-specific and neither is better in the abstract — the question is whether you are short of hands or short of a process.
A managed BPO is a clean answer when a process is settled and you would rather buy the result than run it. Choose Connext when you want the result and the control, at the same cost base.
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.