I was reading do Carmo just now and there he proves that the catenoid is locally isometric to the helicoid, but I wondered why it couldn't be extended to a global isometry and I couldn't figure it out. I'm sure it must be something relatively obvious that I'm missing, so I'd appreciate some help.
EDIT: I think the following argument can be given (but I'm not really sure): we know that intrinsic distance (i.e the infimum of the length of all paths connecting two points) is invariant by isometries and is always at least $\|p - q\|$ (viewing points on the surface as points in $\mathbb{R}^3$). But the intrinsic distance $Y(0,0)=(0,0,0)$ and $Y(0,3\pi) = (0,0,3\pi)$ is evidently $3\pi$, while (background: parameterizing on $[0,2\pi) \times \mathbb{R}$ the catenoid by $X(u,v)=(\cosh(v)\cos(u), \cosh(v)\sin(u), v)$ and the helicoid by $Y(u,v) = (\sinh(v)\cos(u), \sinh(v)\sin(u), v)$) the intrinsic distance of $X(0,0)$ and $X(0,3\pi)$ is greater than or equal to $\sqrt{2 + 9\pi^2} > 3\pi$. We conclude that there exists no global isometry between the catenoid and the helicoid.