Sensitive: Registering very slight differences or changes of condition. Used of an instrument. Sensibility: refers to "the ability to feel or perceive" and a machine cannot do that.

Sensibility is wisdom in your choices. Sensitivity can mean a variety of things. You can be sensitive to pollen which causes your to have hay fever. You can detect the emotional feelings of others. You are sensitive to others needs. You could be easily upset by the comments and looks that you think others are directing at you. You are then said to be sensitive. Some electronic systems can be sensitive to electrical changes. Usually this means they are damaged or perform abnormally. The meanings above for sensitivity are not exhaustive, there are others. Since you want to know the difference between the two words, I think that the meaning associated with emotions is the meaning that you want for sensitivity.