Analogies between VNP and NP
The basic idea is that summing over all Boolean strings (VNP) is like counting the solutions to an NP problem. Even from this perspective, one sees that VNP is more like #P than NP. This is also true as permanent is complete for both VNP and #P. Indeed, the Boolean part of VNP is essentially just #P/poly (it contains #P/poly and is contained in $\mathsf{FP}^{\mathsf{\# P}}/poly$ [Burgisser].

Valiant already observed that $\mathsf{P/poly} \neq \mathsf{NP/poly}$ implies $\mathsf{VP} \neq \mathsf{VNP}$ over $\mathbb{F}_2$, so separating VP from VNP is necessary on the way to proving that NP is not in P/poly. Not only is it a formally necessary first step, but because of the connection between VNP and counting solutions to NP problems, one could imagine that a better understanding of VNP would yield some insight into NP as well.