[howLongHowLong]

Yes, but they're not just from the same root, they're different forms of the same word. Ones the noun, ones the verb. They have exactly the same meaning (except for their grammatical function) but the connotations are almost opposite.

[jacobolus]

I don’t know much about Latin grammar, but from what I can tell both facultas (whence faculty) and facilitas (whence facility) were standard noun forms of the Latin word facilis. Both of these nouns come to English from Latin via French, and neither was newly generated in English.

In Latin (and thence English) facultas meant ability, while facilitas meant easiness.

The adjective facilis comes from the verb facio (to make or do; from the same PIE root as the English verb 'do'), and meant something that can be done/made; something easy, ready, or quick; or someone friendly, courteous, or compliant.

[howLongHowLong]

Sorry, just saw this. You're absolutely right about the etymology, but generally the suffix "-ity" in English serves to make an abstract noun out of an adjective. So it seems more likely to me that "facility" is derived from the English "facile" than developing alongside it. All that said, its just speculation, and facultas->facility/facilitas->facile would explain the subtle difference in meaning.