Characteristic classes, Möbius strip, and the cylinder

You can use the fact that if you cut the open Möbius strip around center the resulting space is connected.

Any homeomorphism from the Möbius strip to a cylinder will induce an isomorphism on fundamental groups, so if a homeomorphism existed it must send the center of the Möbius strip to a curve homotopic to a circle going around the cylinder, and removing anything homotopic to such a curve from the cylinder disconnects it.

As an alternative to James's answer, you can look at the one-point compactifications. For the cylinder, you get a space homeomorphic to a sphere with two points identified, which has $\Bbb{Z}$ for its fundamental group. For the Möbius strip you get a space homeomorphic to the real projective plane, which has $\Bbb{Z}/2\Bbb{Z}$ for its fundamental group.