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In the comments to this question How a principal bundle and the associated vector bundle determine each other, it was remarked that while there is a bijective correspondence between rank $n$ vector bundles with structure group $GL(n,\mathbb{R})$ and principal $GL(n,\mathbb{R})$ bundles, there are situations in which working with one or the other is not equivalent.

I would like to understand why is this the case.

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Short answer: $GL$ does not exhaust all possible Lie groups. A vector bundle always has an associated principal bundle with fiber $GL$ (or a subgroup). In particular, some Lie groups do not admit a faithful linear representation (see this question on MO). Therefore, they cannot be associated to any vector bundle (in a way, no vector bundle "represents them well").

The other way around, given a vector bundle, one can always find an associated principal bundle, as you probably know. But for many theoretical problems, the vector bundle is the most natural object to work with (sometimes, simply because it is a nice module/vector space). For example, on a vector bundle both the action of the group and the sum of vectors are natural operations. Instead, on a principal bundle, the sum is not natural. For principal bundles associated with vector bundles you can try to inherit the sum from the vector space, but it is not guaranteed that summing two sections you get again a section - can you see why? (think of a $U(1)$-bundle...)

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