what does the "constant voltage with external switches" achieve?
Not sure what you mean by achieve but it controls a load by means of an external switching transistor. The voltage applied to the load is independent of the impedance of the load.
What does adding the op-amp do
It turns the driver into a constant current. Here the voltage applied to the load is varied according to the impedance of the load until a certain current flows. This is what you need when driving a high power LED because the impedance of the load changes as the LED warms up.
Your strip is designed to have a constant 12V applied to it so any impedance changes in the LED is taken care of in the electronics of your strip.