For any countable graph $E$, we investigate the relationship between the
Leavitt path algebra $L_{\C}(E)$ and the graph C*-algebra $C^*(E)$. For graphs
$E$ and $F$, we examine ring homomorphisms, ring *-homomorphisms, algebra
homomorphisms, and algebra *-homomorphisms between $L_{\C}(E)$ and $L_{\C}(F)$.
We prove that in certain situations isomorphisms between $L_{\C}(E)$ and
$L_{\C}(F)$ yield *-isomorphisms between the corresponding C*-algebras $C^*(E)$
and $C^*(F)$. Conversely, we show that *-isomorphisms between $C^*(E)$ and
$C^*(F)$ produce isomorphisms between $L_{\C}(E)$ and $L_{\C}(F)$ in specific
cases. The relationship between Leavitt path algebras and graph C*-algebras is
also explored in the context of Morita equivalence.