What distinguished the Möbius strip from the cylinder as fibre bundles? I'm a physicist trying to understand fibre bundles, and I think I'm pretty happy with the wikipedia definition "a space that is locally a product space, but globally may have a different topological structure". I think I also have a grasp, in the general and tangent cases, on the different objects in the fibre bundle: $(E, \pi, M, F, G)$ taken singularly and abstractly. However, I'm having trouble seeking what tells apart two fibre bundles $E$, the Möbius strip and the cylinder. Both have base $M=S^1$ and fibre $F=$ (a segment), so what's different in the two of them? Is it the projection $\pi$, the structure group $G$ or maybe trivially just the total space $E$?

The direct answer to your question about the difference in the tuple you considered is the following: usually the fibre bundle is not defined as the tuple of objects you wrote. The fiber bundle usually is defined to be just $\pi$, and the other objects are named “total space”, “basis” and “fibre”. But what is fundamental here is the map $\pi$. Indeed in the definition of fibration one has conditions on that particular map. So, yes, the main difference stands in how the map $\pi$ is defined, giving rise to different total spaces even with the same base and the same fiber.
Even topologically, they are different objects: we cannot find an homeomorphism between them. For example notice that one is orientable while the other is not (recalling that orientability of a manifold is a property that is preserved under homeomorphisms; like compactness: if $X$ is compact and $X$ is homeomorphic to $Y$, then $Y$ must be compact as well).
Locally they can be considered the same, but notice that their difference is topological, which means a difference in their global structure. Indeed, this is the same difference there is between a cartesian product of the base and the fiber and the whole structure of fiber bundle.
