It is not obvious that the axial and vector form factors are the same. The dependence of them as functions of the momentum transfer (Q^2) is typically assumed to be of the dipole form, i.e. F=F(0)/(1+Q^2/M^2)^2 where M is the cutoff mass. If, F_V(Q^2) and F_A(Q^2) are normalized one they will have similar behavior at low momentum transfer (meaning low neutrino energy). However, the so-called cut-off masses M_A and M_V are different. Consequently, the behavior at large Q^2 (i.e. for the neutrino energies of a few hundreds of MeVs) should be quite different. Typically, one is using M_A=1.014 GeV and M_V^2=0.71 GeV^2. For the vector form factor it exists also expansions beyond the dipole approximation which I use in my codes. 

The lectures of S. Coleman, ``Soft pions'', at the 1966 Erice summer school (reprinted in ``Aspects of Symmetry'') are recommended reading for this question.

I remember that talk of Sidney Coleman, when he was balancing a cigarett between two fingers of his right hand , and a piece of chalk between two other.

Coherent scattering is defined as the scattering on a nucleus which remains on its ground state. Thus there is no exchange of quantum numbers in the t-channel. Thus it involves a neutrino to neutrino transition with the exchange of a Neutral Current. It has to be even more specific the neutral current component must be isoscalar and measures the total baryon number of the nucleus. Therefore there is only one form factor from the vector current ; see  Sakurai and Urrutia  PR  D11,(1975)159).

      The axial current changes parity and does not leave the nucleus intact for this reason there is only vector contribution.

There is also coherent pion production by neutrinos where only the axial current contributes. This a different process and the reference is Paschos and Schalla

PR D80(2009) 033005, with many other references given in this article.