I think it is really natural to believe, after doing Riemannian geometry for a little time, that sectional curvature encodes the all local geometry of a Riemannian manifold. One of the first thing one learns is that having constant sectional curvature implies that you are locally isometric to either the sphere, the euclidian space or the hyperbolic space.

The paper http://www.jstor.org/stable/1970580 gives a satisfying answer to the question in dimension higher than 4 (Any diffeomorphism preserving the sectional curvature is an isometry, except in the constant case, which we already know to be locally determined by the curvature). (See also the MO question : Determining a surface in $\mathbb{R}^3$ by its Gaussian curvature for an interesting discussion on the question.)

Can anyone give a example of two smooth metric on a surface having the same Gaussian curvature function ?

One could say : take two different hyperbolic compact surfaces of the same genus, their curvature functions are both -1, and yet are not globally isometric. But still those surfaces are locally isometric.

So can one find two metric $g_1$ and $g_2$ on the disk inducing the same curvature function such that no diffeomorphism of the disk induces an isometry on a neighborhood $(U_1,g_1)$ of $0$ on $(U_2=f(U_1),g_2)$. Which are essentially different is this sense.