In any case where parties do not agree on a solution, communication of concerns and desired outcomes is key. Each person is the center of their own universe and needs to communicate that viewpoint.
This means that authentic behavior is critical to actually understanding the differences so that satisfactory resolution is possible.
One must be honest and open about concerns. All too often the expressed concerns simply camouflage a deep-seated, underlying concern. This can be at a sub-conscious level, so that the person in question is also not aware of the true reasons underlying their behavior.
This means that one should be open enough to question one's own motivation. Is your response appropriate, or does it seem excessive or unreasonable considering the actual issues involved. If it seems unreasonable, then it may well mean that it is based on a deep-seated, unreasoning fear from past experience. One should try to make an impartial decision based on the facts at hand, keeping this bias in mind.
This demands emotional maturity and a depth of personal insight, but both of these are needed to resolve many conflict situations.
If one party admits thet perhaps their reactions are based on some purely understood underlying cause and not the facts currently in evidence, it will often trigger a similar response from the other party. Such openness and authentic behaviour exerts moral pressure on the other to do likewise.
Perhaps one way of overcoming the differences is to ask how an impartial bystander would view the conflict? Ask the question, and let both parties discuss it might appear to a bystander. This helps remove the personal element. If the argument would seem ridiculous to a bystander and yet seems so critical to the participants, there is clearly some other motivation involved apart from that on the surface.
it is important to remain objective in order to achieve resolution of a conflict, and this "bystander" approach helps to highlight the objective element.