Why use convolutional NNs for a visual inspection task over classic CV template matching?
The answer depends on the task.  Template matching can work for some tasks but not for all.  CNNs potentially have the ability to generalize to unseen inputs that don't match any of your templates, so can potentially generalize better.

But whether CNNs will beat template matching will depend on the specific task and what specifically you're trying to achieve.  This is an empirical science; ultimately, the way you find out which works better is to try them both -- or learn from others who have tried them (e.g., by reading the literature).  I don't think you're going to find some theory or taxonomy that is going to substitute for empirical evaluation on real-world data.