The most useful feature in math programming is the optimality bound estimate it offers.

It is very important to know that every single time your solution is at worst 1% away from the proven optimal one. If it is not, you get a signal and you can invoke mitigations.

This is when pure heuristics fail. They may work 99% of the time giving you near optimal solutions, but that 1% they will fail you and even worse, you will have no idea that they failed you.

Approximation algorithms also offer some bounds, but typically they are very loose, to the point that they are not useful to the bean counters. Nobody is running their fleet using (exclusively) the Christofides algorithm.