Awareness is akin to long, deep, and deliberate breaths; wouldn't you agree? It can also be akin to short, shallow, and accidental breaths; would you not agree?
Sensitivity is akin to unstructured, unknowing, and reactive breaths; wouldn't you agree? It can also be akin to structure, knowing, and active breaths; would you not agree?
So, awareness and sensitivity aren't exactly opposites. What they have in common is they can affect just about any situation. What is the difference? Awareness is akin to relaxation. Sensitivity is akin to tension. This is the only way in which the two differ that i know of.
Being aware implies a relaxed understanding. Being sensitive implies a tense understanding. Awareness signifies internal peace. Sensitivity implies internal tension.
If sensitivity is an issue internally, one must look for the tensors in ones life. Dealing with these, one can be at ease and relax into their awareness, letting go of the sensitivity caused by the tension.
This may help some meditate deeper, and with greater ease.
May this be helpful as it can be.