Number of Vertex Covers and Permanent
If $G$ has an isolated node, the permanent will be equal to 0.

If you know that permanent is 0, it tells extremely little about the number of vertex covers in $G$. You can take any graph $G$ and construct $G'$ by adding an isolated node; the number of vertex covers in $G'$ is exactly 2 times the number of vertex covers in $G$ – which can be virtually anything – while the permanent of $G'$ is 0.

Conversely, if you know the number of vertex covers, you can't tell if the permanent is 0 or not. For example, you can take any graph $G$ with a non-zero permanent and construct two new graphs: $G_1$ is $G$ + a triangle, and $G_2$ is $G$ + two isolated nodes. Both $G_1$ and $G_2$ have exactly 4 times as many vertex covers as $G$. However, $G_1$ will have a non-zero permanent while $G_2$ will have permanent 0.