Having grown up with one parent a militant atheist, and the other a new-age devotee, I've sort of been threading this needle my whole life.  From what I've seen, much of the hostility comes from a basic epistemological incompatibility that is poorly understood on both ends, and that it is the failure to understand this basic logical and evidential problem that leads to so much of the hostility.

In essence, both sides are talking past each other, but the problem is that they think it's possible to talk to each other, and get angry when they are misunderstood - when in fact, understanding is fundamentally impossible.