The associated family of a minimal surface gives a tangible counterexample. The Weierstrass representation lets you cook up a conformally parameterized minimal surface from a meromorphic pair $f \sqrt{dz}$, $g \sqrt{dz}$.
The parameterization is then given by $$F(x,y) = Re\int_0^{x+iy} (f^2 - g^2, i(f^2 + g^2), 2 fg) ~dz$$
The normal map of $F$ can be obtained by thinking of $g/f$ as a map to the Riemann sphere, and the metric induced by $F$ is just $4(|f|^2 + |g|^2)^2 |dz|^2$. From this data, you can cook up the Gauss and mean curvatures, and it happens to be true that if $f \sqrt{dz}$ and $g \sqrt{dz}$ are meromorphic, you get a minimal surface.
But then consider what happens if you multiply both $f$ and $g$ by $e^{i \theta}$ --- the normal map and the metric are both unchanged, and $e^{i\theta} f \sqrt{dz}, e^{i\theta} g \sqrt{dz}$ are still quite meromorphic, so you get a new minimal surface which is isometric to your old one. This means you have made a new surface whose principle principal curvatures agree with your old one!
I think the moral here is that even knowing the metric and the complete set of principle principal curvatures isn't enough to reconstruct a surface --- the curvature directions are also vital data.
To see all this in action, here is a video with strange music showing the helicoid transforming into the catenoid, which starts with the Weierstrass data for the catenoid and then multiplies by $e^{i \theta}$, with $\theta$ increasing as the movie progresses. Every one of the surfaces is isometric to the catenoid! But they do have different second fundamental forms.
