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I was wondering is there any difference between them? As all fibers are fiber bundles, so surely the fibers are the same. But, then couldn't there be some special thing about a fiber of fiber bundle that isn't in covering spaces.

As Hatcher doesn't define fiber when he is describing what a fiber bundle is.

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A "fiber" means the pre-image of a point. A "fiber bundle" is roughly a map $X \to Y$ such that the pre-image of every point $y \in Y$ looks the same independent of $y$ (precise definition in Hatcher). When talking about a fiber bundle, we often say "the fiber is (adjective)" or "the fiber is (particular topological space)" to mean that any fiber has this property. A covering space is a fiber bundle where the fiber is discrete.

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