Is there an "intrinsic" difference between a plane and a cylinder? Since the plane and the cylinder have zero Gaussian curvature, I'm wondering, is there an "intrinsic" way of telling one from the other?
By "intrinsic" here I loosely mean a property that can be calculated and/or deduced by inhabitants of the manifold itself, without "seeing" it from a higher dimensional space.
 A: A resident of cylinderland would find that at each point, there are two distinguished directions to shine a laser pointer where the pointer (eventually) illuminates itself.  Equivalently, along this distinguished axis, this universe appears to be periodic.
Quantum fields on cylinderland would have discrete excitations associated with this compact dimension.
A: Hint:
not the curvature, but the existence of closed paths (loops) that cannot be reduced to a point by a continuous transformation. 
A: $\pi :{\bf R}^2\rightarrow C:={\bf R}^2/{\bf Z}$ is a universal covering where $C$ is cylinder. Since $C$ is quotient of ${\bf Z}$-isometric action, so $\pi$ is local isometry. Hence locally same but different fundamental group
(In further (1) they have different volume growth $f(R):={\rm vol}\ B(x_0,R)$ and (2) In ${\bf R}^2$ any two points in any ball $B(x_0,R)$ can be connected by line segment in it but if we have suitable $B(x_0,R)$ in cylinder, this can not happen.) 
A: If you define lengths (and angles of curved surface that can be computed out of it) then there is no difference between a cone, a cylinder or other developable of zero $K$ Gauss double curvature (isometric equivalents). $K=\frac {LN-M^2}{EG-F^2}$, comprises of  coefficients of first/second fundamental forms in Classical surface theory is  entirely determinable by Gauss Egregium theorem from the first form alone. It remains the same in Bendings of surface without tearing apart points in the fabric / membrane of space-time
FlatLanders in relativity theory (only parts I remember from  book Differential Geometry & Relativity Theory: An Introduction by R.L. Faber 1983  CRC Press) can get information about surrounding space even if they  are third (normal) dimension "blind" , or unable  to compute $( L,M,N)$  directly. Without using telescope, gyroscope GPS or other external inputs how that is possible I  too wonder, would like to know.
A: Locally there is no difference, but globally there is a difference.
Pick a point on the cylinder; look at a small neighborhood of that point.  One can unroll the neighborhood and lay it flat in a plane without stretching the surface, so all distances in the plane are the same as the corresponding intrinsic distances on the cylinder.  So you can't tell the difference.
However, if you go far enough in a particular direction on the cylinder, you'll return to where you started.  That doesn't happen in the plane.
A: Algebraic topology is the answer to such questions. In short, the first homotopy groups of these spaces are different. Every closed loop in the plane can be continuously contracted to a point, but not so for some loops in the cylinder.
See my answer here for what it might be like to live in a cylindrical space.
The intrinsic property that the plane has that the cylinder does not is simple connectedness.
A: Whenever one considers vectors on a surface S based at the point p, one consider this vectors as elements of tangent space. As a matter of fact the tangent space to the surface S at p exists if $X^{-1}(p) = q$, where $X$ is a parametrization of S, the differential $dX_q$ exists, and $dX_q$ has maximal rank. For a cylinder a natural direct product on the tangent space is the same that for a plane. This is because a plane and a cylinder shares many properties and locallyyou cannot distinguish them. As a matter of fact they have the same first fundamental form. Of course, there is no doubt that a plane and a cylinder are different surfaces. When looking on how the normal vector evolves on a cylinder in comparison to how it evolves on a plane you can understand the difference. Cylinder and plane have different second fundamental forms.
