In the former case we are taking data, labeling it, then using it to build our nets and models. You are correct to an extent that it's a usable model once trained and that data is less important.

However, equally if not more important is the data that is being put into the net to come out as a result/action. Arguably this data comes through the same pipe as training data - and the pipes are similarly limited. So its ALWAYS important because you can't take an action or classify or otherwise without it.

When you add in the reinforcement mechanism, or later unsupervised techniques then those data mechanisms blur between training and action data so the point is moot. It's not a one run process in the long run, it's iterative and always evolving based on the user.