Are cylinder and cone diffeomorphic?

Question

I'm a beginner to differential geometry. I proved the cylinder $\{(x,y,z)\in \mathbb{R}^3: x^2+y^2 = a\}$ for some $a>0$ and the cone $\{(x,y,z)\in \mathbb{R}^3: x^2+y^2 = z^2, z>0\}$ are regular surfaces. I think of some mappings between and see that they are noninvertible at some point. Then I try to prove they are non-diffeomorphic, but cannot see how to utilize the observation.