[question] What's the difference between two words: faculty and facility?

Facility : She has great facility in playing the piano.

Faculty : He has a faculty for putting people at ease.
According to the dictionary I have,
both of the two words above seem to have similar meanings talent, ability, or skill.
Is there any delicate differences between two words?

If so, please let me be understood.
For instance,
a) He has a facility for putitng people at ease.
b) She has great faculty in playing the piano.
You think, as a native speaker, these two sentences are pretty akward or something? Or... the meaning has been changed somewhat compared to the ones on the top?
What a mystery, please help me.