To me, if a question says "show that ..." that means I take some starting point as a given (for example, one of the Maxwell equations for EM) and apply some processes to it until I get the result asked for. This may just be a single line of calculations over several steps or it may require some natural language explanation of the methods, or references to the symmetry of the problem to simplify it. I was trained a physicist though, so obviously that's a bit different. I never did any proofs but I assume you would start by stating axioms/lemmas/corollaries etc and every step must follow with infallible logic from one to the next until the result is obtained. I suppose from my perspective the different could be simplified by saying 'show that' is an exercise in equation manipulation whereas 'prove that' is an exercise in logic.
We were never asked to prove anything as physics students because nothing in physics has been proven as everything relies on the way we perceive the world around us which is not necessarily true. Even in the maths courses we did the lecturers weren't allowed to ask "prove that" questions because we lacked the skills/vocabulary to do it properly, even for purely mathematical questions with no basis in physics.