If we begin with the assumption that politicians are selfless servants of the people who only want to do what's best for society at large in the long run, then you would conclude that politicians will carefully examine the theories and methods of the two economists' schools of thought. If we begin with the assumption that everyone, including politicians, is primarily self-interested, then you would conclude that a politician would just do what's popular with no regard for how the two economists arrived at their advice.
I believe people are primarily self-interested. From that perspective none of this discussion about economic theories matters because politicians are just going to do what's popular anyway.
The proposition that people are primarily self-interested is therefore in my opinion the only idea of value that's come out of the study of economics, which is the study of human behavior after all.