Facilities" is a silly word that can refer to just about anything. You have "facilities" at a gym (work-out equipment, etc.), at a university (library, cultural center, etc.), just about anywhere. Even prisons are sometimes called "facilities," as are various types of buildings ("What a nice facility!"). Basically, I would avoid using this word and opt for a more precise one, unless you actually wish to say "facility" as in "ease" ("I admire the facility with which you learn languages," for example).
"Amenities" (note the spelling) are "services." This word cannot be used to describe buildings. When you rent an apartment, some of the amenities that can come with it are electricity, water, heating, etc.