Let $G$ and $H$ be two groups, and $f$ a map from $G$ to $H$ ($\forall g\in G \Rightarrow f(g)\in H$). Then $f$ is a homomorphism if $\forall g_1,g_2\in G \Rightarrow f(g_1g_2)=f(g_1)f(g_2)$. This means that $G$ and $H$ are algebraically identical.
Isomorphism is a bijective homomorphism.
I see that isomorphism is more than homomorphism, but I don't really understand its power. When we hear about bijection, the first thing that comes to mind is topological homeomorphism, but here we are talking about algebraic structures, and topological spaces are not algebraic structures.
Then what is the "power" that makes us to define isomorphism as a special case of homomorphism?