ML for classification vs prediction and dataset labeling

In the ML research community, is there clear distinction between ML applied for future prediction vs classification? (specifically, as it pertains to datasets with self-evident labels)
It seems an important difference is that ML applied to future prediction (e.g. predicting weather patterns, stock market fluctuations) can generate automatically labelled datasets. That is, the classifications of interest (humidity increases, stock prices drop…) may be self-evident in the dataset and would not require human labeling. Alternatively, ML used for classification (e.g. Is the user running? Is that object a car?) require human labelling.
It seems that understanding this difference could have important implications in product development. If my ML product focuses future prediction with a self-evident outcome, it can train itself without a user needing to label inputs. As a bit of a silly example, my cat’s collar could learn to recognize the movement patterns predictive of her running away (perhaps defined by moving a >30 m from home) and my app will warn me (e.g. >0.8 probability of eloping) if I need to keep a better eye on her.
What have other people have thought or read about this distinction?