You seem to be confusing two completely different concepts. You can use "cylindrical coordinates" (no need for the word "polar"- cylindrical coordinates are necessarily polar coordinates with the z-axis added) for many different kinds of problems, not just finding surface area. On the other hand, it is certainly possible to find surface area of a cylinder in Cartesian or even spherical coordinates.
If the problem is to find the surface area of a cylinder then it depends on exactly what the question asks. Typically, if you are given a cylinder with a specificed length the surface area includes the two end points, while the "curved surface" does not. Of course, if you are given just an equation for the curved surface, that does not include the endpoints.