This is too long for a comment, but it is one to this question:
I am reading first chapter in that book and they conclude with idea of
Universal bundle $E_n\rightarrow G_n$ with the property that all n
dimensional bundles over paracompact spaces(we are not on smooth
manifolds, we are working only on topological spaces) are obtainable
as pullbacks of this single bundle. They say nothing about
classification of vector bundles on manifolds, classification of
vector bundles on smooth manifolds, classification of principal g
bundles on Smooth manifolds. Can you suggest some reference or should
I ask it as a separate question.
There is essentially no difference: Let's call a vector bundle continuous if the transition maps are continuous, and call it smooth if the transition maps are smooth.
Since every manifold is paracompact the continuous vector bundles are clearly classified by the maps you mention. Then the question becomes if two smooth vectorbundles are isomorphic as continuous bundles, they are isomorphic as smooth bundles: This is true. The bundle $E_n\rightarrow G_n$ is smooth, and smooth bundles are classified by smooth homotopy classes of maps into $G_n$. But any two (smooth) maps which are homotopic as continuous maps are smoothly homotopic. This follows from the fact that any continuous map can be approximated by a smooth one.
About the principal $G$ bundles: I think there is a universal principal $G$ bundle over $BG$ and any principal $G$ bundle arises as the pullback of this bundle along a classifying map. A reference for this would be Husemöller's Fiber bundles if I remember correctly.