Usefulness of the concept of equivalent representations Definition: Let $G$ be a group, $\rho : G\rightarrow GL(V)$ and $\rho' : G\rightarrow GL(V')$ be two representations of G. We say that $\rho$ and $\rho'$ are $equivalent$ (or isomorphic) if $\exists \space T:V\rightarrow V'$ linear isomorphism such that $T{\rho_g}={\rho'_g}T\space \forall g\epsilon G$. 
But I don't understand why this concept is useful. If two groups $H,H'$ are isomorphic, then we can translate any algebraic property of $H$ into $H'$ via the isomorphism. But I don't see how a property of $\rho$ can be translated to similar property of $\rho'$. Nor I have seen any example in any textbook where this concept is used. Can someone explain its importance?

Specific representations of groups can give us concrete information about them. Thus, much like groups themselves, we wish to classify them. To do this we need a notion of "equivalence" of representations, and this is exactly the right notion.
Without it, we are left grappling with an infinite number of representations on isomorphic vector spaces. Fortunately in nice situations, an arbitrary representation will decompose into a finite sum of a finite number of different (note we need the concept of equivalence to define a genuine notion of "diference") irreducible representations.
A similar example in group theory is that if we wish to understand a product $G \times H$, it really suffices to just understand $G$ and $H$ individually. This is what happens in the direct sum decomposition of representations.
