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*the difference between science and mathematics.

Physicists normally care little whether their mathematical tools are formally proven correct. See: the Feynman path integral. It makes sense and gives correct answers, but as far as I know, it has no rigorous mathematical basis.

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Indeed, natural science deals in models, not 'the truth'.

In most cases 'correct' isn't an option, rather a degree of accuracy. If it's consistent with the measurements, or even better if it has predictive power, then it's useful.

I wouldn't say they don't deal in truth. Scientific theories clearly capture a lot of true elements of how the universe works. But proving that the mathematical methods are correct, starting from basic axioms, is something for mathematicians. Most physicists will use a mathematical tool if it's useful, even if it's not been formally proven correct.

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