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I assume that by $\mathbb{R}^n$-fiber bundles you mean differentiable bundles with fiber diffeomorphic to $\mathbb{R}^n$. So you are forgetting the linear structure of $\mathbb{R}^n$, keeping only the smooth structure. It is a fact that $\textit{Diff}(\mathbb{R}^n)$ deformation retracts to $\textit{GL}(n,\mathbb{R})$, hence yes, the classification of rank $n$ real vector bundles is essentially the same as that of $\mathbb{R}^n$-fiber bundles. I don't know whether this holds at the topological level already (that is, whether $\textit{Homeo}(\mathbb{R}^n)$ deformation retracts to $\textit{GL}(n,\mathbb{R})$).