Sensitivity = responsiveness, reactivity, the condition of being sensitive.

• The eyes are sensitive to light.
• Her skin is easily hurt because of its high sensitivity to the sunshine.
• As an author he should not have so much of sensitivity to criticism.
• When it comes to religious matters people in India tend to have more sensitivity.
• Mammootty has the sensibility and sensitivity as to how to deal with his characters.

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Sensibility = feelings, discernment;  the ability to decide what is good or valuable, especially in connection with artistic or social activities.

• We Indians show respect for the sensibilities of others.
• She is a woman of poetic sensibility.
• The study of literature leads to the growth of intelligence and sensibility.
• We have to be careful not to offend our readers’ sensibilities.
• It is not sensible to spend so much on weddings. ( i.e not rational, reasonable ,or logical etc)
• You can rely on his advice.  He is a sensible man.
• That was a very sensible decision.