My favorite such functor is the forgetful functor from "$C^\infty$ manifolds with real coefficients" (defined as spaces with a sheaf of $\mathbb{R}$-algebras locally isomorphic to the sheaf $C^\infty_\mathbb{R}$ on the space $\mathbb{R}^n$) to $C^\infty$ manifolds with complex coefficients (define as spaces with a sheaf of $\mathbb{C}$-algebras locally isomorphic to the sheaf $C^\infty_{\mathbb{C}}$ on $\mathbb{R}^n$). This equivalence fails to hold in the supergeometric world (as well as in various other extensions of the theory of real manifolds).