Staff augmentation
Staff aug gets you a person. The rest, sourcing, vetting, training, skill development, day-to-day management, lands on you. Depending on your team size, that person may work alone, with no one to pair with, review their code, or catch what they miss. Whatever they can't handle comes back to your desk to solve.
Working with a traditional agency
Agencies start with a scope of work, and that document becomes your job. Features get negotiated upfront, changes get papered, and you sign a long commitment before you fully know what you need. The deeper cost shows up later: you handed over ownership of what gets built.