In many Western cultures, particularly American culture, disagreement can be seen as a healthy and direct part of communication (“let's agree to disagree”). However, in Chinese culture, which places a high value on group 和谐 (héxié) - harmony, a 分歧 (fēnqí) is often viewed more seriously. A 分歧 is a disruption to this harmony. It's a divergence that, if not handled carefully and often indirectly, can lead to a loss of 面子 (miànzi) - “face” or social prestige - for the involved parties. Therefore, addressing a 分歧 is not about winning an argument, but about finding a way to 解决分歧 (jiějué fēnqí) - “resolve the disagreement” - or 弥合分歧 (míhé fēnqí) - “bridge the divergence” - in a way that allows the relationship and forward progress to continue. This often involves compromise, mediation, or one party conceding for the greater good of the group or project. The goal is to bring the diverging paths back together.
分歧 (fēnqí) is a relatively formal word. You wouldn't use it for a trivial spat with a friend, but it's very common in more structured or serious situations.