There are various configurations of breakout boards but generally, they consist of a socket on a PCB where the pins are fanned out to a series of test points.  The test points maybe 1, 2, or 3 pin jumper systems, banana jacks, or other types of test points. The primary limitation is that the socket and board are fixed and cannot be reconfigured.
The switch box decouples the test points from the socket and DUT board. The DUT board can be swapped for any other component footprint and the total number of test points can be expanded by adding more switch boxes.
A switch box and DUT board is generally a lower cost solution because the switch box is a standard component and the DUT board is simpler than a breakout board.  A breakout board can be designed with special features and interface to your cables or work with daughter cards. In some cases, the breakout board will be a better solution for technical reasons.