Basically the same meaning in everyday english Yoong.
Some might argue that amenities has connotations of more luxury. where facilities could be less luxurious.
The difference between a 5 star hotel bathroom and a public restroom is an example.
facility
Something designed, built, installed, etc., to serve a specific function affording a convenience or service: transportation facilities; educational facilities; a new research facility.
6. Often, facilities. Informal. a rest room, esp. one for use by the public, as in a theater or restaurant.
http://dictionary.reference.com/browse/facility
Note the lack of reference to pleasure and comfort in the definition of facility
Note also that there can be differences in the meaning of facility (readiness or ease due to skill or aptitude) He had a great facility for auto repairs.
Amenity
2. any feature that provides comfort, convenience, or pleasure: The house has a swimming pool, two fireplaces, and other amenities.
3. the quality of being pleasing or agreeable in situation, prospect, disposition, etc.; pleasantness: the amenity of the Caribbean climate.
4. amenities, lavatory; bathroom: used as a euphemism.
Note the use of "pleasure" and "comfort" in the definition of amenity.
http://dictionary.reference.com/browse/Amenity