Is there an "intrinsic" difference between a plane and a cylinder?

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.

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.