The two models look similar on a proposal and behave very differently in practice. The difference is not the people. It is where responsibility sits.
Staff augmentation
Engineers join your team and work to your process. You set the priorities, run the standups, review the code and decide what done means. The engineers bring skills; you bring direction.
This works when you already have a functioning team and a clear backlog, and the constraint is capacity or a specific skill you do not have in house.
A managed team
A team takes a defined outcome and is accountable for delivering it, including its own planning, quality and release process. You review progress against the outcome rather than managing day to day.
This works when you do not have the engineering management capacity to absorb more people, or when the work is separable enough to hand over cleanly.
The mistake that costs most
Buying augmentation when you needed a managed team. Engineers arrive, nobody has time to direct them, and the work drifts. The engineers are blamed for a gap in direction that was never theirs to fill.
The reverse happens too: handing over an outcome that is tangled through the rest of your system, so the team cannot move without constant access to yours.
A practical test
Can you name the next six weeks of work for an extra engineer today? If yes, augmentation fits.
Would an extra engineer need more of your lead's time than they free up? If yes, you need a managed team.
Can the work be described as an outcome rather than a task list? If yes, a managed team can own it.
Does the work touch systems only your team understands? If yes, keep ownership in house and augment.
Neither model is better. They answer different questions, and the engagement goes wrong when the question was never asked.