Difference between "Show" and "Prove"
Sometimes students misinterpret show to mean give an example. I now avoid using show in exams; I always use prove when a proof is required.
In the context of examples or calculations, it might be ok use show. For instance, "Show that $2$ is a root of $x^2-4$" or "Show that $\sin x$ is a solution of $y''= -y$.
No difference. Just a way to fool students...
No difference. Just another way of stating a question.