a physicist is openly willing to compromise their principles and stretch the truth for the sake of press, while a mathematician sticks to the truth and as a result nobody cares.I think it goes deeper than this. I was in the same undergraduate mathematics and physics classes as Nima Arkani-Hamed[1]. In calculus class, we learned very formal, epsilon-delta calculus (from Spivak's textbook). In physics, we learned from Feynman's Lectures on Physics.
Years later I went to lunch with him, and he mentioned that he spent much of his time fighting the formalist instincts from our Calculus class, and thinking more intuitively about mathematics. I do think physicists play more fast-and-loose with mathematics, e.g. using approximations like a few terms from a Taylor series, or other assumptions. When talking to the public, I can see how that would turn into playing fast-and-loose with the facts in order to "project" their knowledge into the space spanned by laypeople's knowledge, even if that projection loses important aspects of the work.
[1] https://en.wikipedia.org/wiki/Nima_Arkani-Hamed