In your mind, what is the distinction between the two?

I don't see any way to draw a strong distinction between what I did during my physics days and what I do today. Different math, sure - PDEs vs statistics. But then again the math changes when you go from quantum physics to statistical mechanics also.

You tell things to your hairdresser, and you tell things to your priest, but that doesn't mean your hairdresser is bound by the conventions around the confessional, does it? So you did some maths for a guy and signed an NDA, don't make it something that it's not.

Um, you dodged the question. I didn't ask "what's the distinction between a hairdresser and priest".

Umm, not at all. You are not a "muzzled scientist", a million of us have signed NDAs without making a big song and dance about it.

The question which you have failed to answer is "what is the distinction between data scientist and scientist"?