Relation/Difference between moduli spaces and classifying spaces.
The main difference is that maps to moduli spaces represents certain classes of maps to an object in the same category, while in classifying spaces they are in different categories: maps to the classifying space are defined only up to homotopy, i.e. maps in the homotopy category, while vector bundles are defined in the category of topolgical spaces - any vector bundle is homotopy equivalent to the original space. See here for more details: https://ncatlab.org/nlab/show/moduli+space#because
This is only a partial answer but too long for a comment.
At least for principal $G$-bundles, any model for the classifying space $BG$ is a "space of $G$-torsors". By "$G$-torsor" I mean a topological space with a free and transitive $G$-action, for example the fibres of a principal $G$-bundle.
There is a topological characterization of $BG$ as follows:
Suppose $E$ is a contractible space with a free $G$ action such that the quotient map $E\to E/G$ is a fibre bundle. Then $E \to E/G$ is a model for the universal principal $G$ bundle. (In particular $E/G$ is a model for $BG$.)
Moreover every universal bundle $EG \to BG$ arises in this way.
But what is the space $E/G$? Each point in $E/G$ is a $G$-orbit in $E$, which is already a $G$-torsor. Any continuous function $f\colon X \to BG$ picks for each $x\in X$ a $G$-torsor $f(x)\in BG$, each already equipped with a $G$-action from $E$, and because $f$ is continuous these actions also vary continuously from fibre to fibre resulting in a principal $G$-bundle over $X$. Varying $f$ by a homotopy results in a different but isomorphic principal bundle.
In certain cases our group $G$ is the structure group of a different type of bundle we're studying: for example $O(n)$ is the structure group for rank $n$ vector bundles, and if $M$ is a smooth manifold $Diff(M)$ is the structure group for $M$-bundles. In special cases the classifying space can be modelled using moduli spaces of these fibre types: $BO(n)$ can be described as the Grassmannian $Gr_n(\mathbb{R}^\infty)$ of all $n$-dimensional linear subspaces of $\mathbb{R}^\infty$, where $O(n)$ has a free transitive action on the contractible Stiefel manifold $St_n(\mathbb{R}^\infty)$ of $n$-frames, and $BDiff(M)$ the moduli space of submanifolds of $\mathbb{R}^\infty$ diffeomorphic to $M$, where $Diff(M)$ acts on the space of embeddings $Emb(M, \mathbb{R}^\infty)$. (Note that these are only really classifying spaces for bundles over paracompact spaces.) In these cases we are able to identify each $G$-orbit with the fibre type we're interested in.
I have often wondered if for any $G$ and any $G$-space $F$ whether we can model $BG$ as a moduli space of objects of "type" $F$ as in the case of vector and manifold bundles, but I do not know.