Denote the line connecting the centers by $\ell$ and let $K,L,Q,R$ be midpoints of segments $AC,BD,AB,CD$ respectively; obviously points $Q,R$ lie in a radical plane of spheres, so easy to see, that $P$ is a projection of these two points on $\ell$. Also $\overrightarrow{KQ} =\frac{1}{2}$ $\overrightarrow{CB} =\overrightarrow{RL}$, and thus $|PM|=|KQ|\cdot cos\angle (\overrightarrow{KQ} ;\ell )=|RL|\cdot cos\angle (\overrightarrow{RL} ;\ell )=|PN|$ so we are done.