When do we have $C(G) \otimes C(G) =C(G\times G)?$

Question

Let $G$ be a group and $C(G)$ be the set of complex valued functions on $G$.

Can someone please let me know why $C(G) \otimes C(G)$ is only dense in $C(G\times G)$ and the two are different if $G$ is not a finite group.

The idea comes from defining a Hopf algebra structure on $C(G)$ and we need to define the co-multiplication $\Delta: C(G) \to C(G) \otimes C(G) $!

Thanks!