I think of bundles as being a generalization of the product. When using a product I can construct new spaces. But the definition of a bundle as a triple $(E, \pi, B)$ requires that I already have a total space $E$ ready. This makes me think that a bundle isn't really a tool for creating new spaces, but for analyze them.
As a quick example, I can construct a möbius strip using quotient spaces, and later on analyze it as a bundle.
Am I right to think of it this way?
For better comparison, one can describe products in the same language: a product is a collection $(P,\pi_1,\pi_2,X,Y)$, where $\pi_1:P\to X$ and $\pi_2:P\to Y$ are maps satisfying certain axioms. In more category-theoretic settings, this is the standard definition. The key property of products is that the entire collection is determined (up to isomorphism) by $(X,Y)$ alone. For bundles this is not the case: we clearly cannot determine a bundle from the base space alone, and even if we specify the base space and fiber, some additional topological data is needed to uniquely determine the bundle.
That said, this is a somewhat limited view of the role that products play, and even more so for fiber bundles. A bundle isn't just the total space $E$, it is the whole structure $(E,\pi,B)$. They aren't used to analyze spaces which happen to be the total space $E$ of some bundle; rather, they arise naturally when one tries to assign some kind of "local data" to a space, which happens all over topology and geometry: some examples being the tangent bundle of a manifold, the various tensor bundles which arise from it, orientations, covering spaces, and many others besides. Fiber bundles provide a single language with which to describe the common properties of all of these structures.