= Solution
The helicoid has no angular identification: its height coordinate is $v$. Consequently
$$
\int_S K\,dA
=-\int_0^\infty\int_0^\infty
\operatorname{sech}^2u\,dv\,du
=-\infty.
$$
The catenoid has total curvature $-2\pi$. A global <Riemannian isometry> preserves both the Gaussian curvature and the area element, hence preserves total curvature. Therefore
$$
\boxed{S\text{ and }S'\text{ are not globally isometric}.}
$$
The same obstruction is visible in the parameter map: increasing $v$ by $2\pi$ gives a different point of the helicoid but the same point of the catenoid.
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