As the measure of the closeness of two distributions $p_A$ and $p_B$ You could use the <A HREF="https://en.wikipedia.org/wiki/Bhattacharyya_distance">Bhattacharyya coefficient</A>
$$w=\int \sqrt{p_A(x)p_B(x)}\,dx\in[0,1],$$
which for two Gaussian distributions (means $a,b$; variances $s^2$, $m^2$) is given by $w=e^{-d}$ with
$$d=\frac{1}{4} \ln \left [ \frac 1 4 \left( \frac{s^2}{m^2}+\frac{m^2}{s^2}+2\right ) \right ] +\frac{1}{4} \frac{(a-b)^{2}}{s^2+m^2}. $$