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?

This is just the concept of isomorphism applied to representations, i.e. $T$ is providing an isomorphism between $V$ and $V'$ which interchanges the two group
representations. So all your intuitions for the role of isomorphisms in group theory should carry over.  
Why are you having trouble seeing that properties of $\rho$ can be carried over to
$\rho'$?  As with any isomorphic situations, any property of $\rho$ that doesn't make specific reference to the names of the elements in $V$ (e.g. being irreducible or not, the number of irreducible constituents, the associated character) will carry over to $\rho'$.
