I can give you an informal reason for why the separation would not prove $P \ne NP$.
VP and VNP focus on algebraic functions which degree is bounded by a polynomial. Notice that it is easy to compute in an algebraic function of exponential degree with a polynomial size algebraic circuit.
There is a well known1 depth reduction for algebraic circuits: any polynomial size algebraic circuit computing a polynomial of degree $d$ can be turned into an algebraic circuit of polynomial size and depth $O(\log d \log n)$.
You may think of $VP$ as an algebraic variant of $NC^2$, thus proving that $VP \ne VNP$ amounts to prove an algebraic non-uniform equivalent of $NC^2 \ne \# P$. That would not rule out $P = NP$, at least not immediately.
Disclaimer: I can't access the paper right now and I don't remember if the reduction works in any field or just in finite ones.
1 L. G. Valiant, S. Skyum, S. Berkowitz, C. Rackoff. Fast parallel computation of polynomials using few processors. SIAM J. Comput. 12(4), pp. 641-644, 1983.