If you say that a situation or problem is of some delicacy, you mean that it is difficult to handle and needs careful and sensitive treatment. There is a matter of some delicacy which I would like to discuss
the quality of being beautiful and delicate in appearance; "the daintiness of her touch"; "the fineness of her features"
If someone handles a difficult situation with delicacy, they handle it very carefully, making sure that nobody is offended. Both countries are behaving with rare delicacy = sensitivity
Delicacy is the quality of being easy to break or harm, and refers especially to people or things that are attractive or graceful. the delicacy of a rose
A delicacy is a rare or expensive food that is considered especially nice to eat. Smoked salmon was considered an expensive delicacy
Nicety or fineness of form, texture, or constitution; softness; elegance; smoothness; tenderness; and hence, frailty or weakness; as, the delicacy of a fiber or a thread; delicacy of a hand or of the human form; delicacy of the skin; delicacy of frame