【Amenity】 and 【luxury】 both denote something (as an object, a feature, a quality, or an experience) that gives refined or exquisite pleasure or is exceedingly pleasing to the mind or senses.

【Amenity】 typically implies a delightful mildness, gentleness, or softness, especially in contrast to an uncomfortable or distressing harshness, roughness, or crudeness.

• many English go to the Riviera in the winter because of the 【amenity】 of its climate

It may imply no more than a vague conducing to physical or material comfort or convenience.

• every 【amenity】 . . . including... showers, central heating, and first-class cuisine
  —H. G. Smith

【Luxury】 stresses keen, often voluptuous, enjoyment and unalloyed gratification of the mind or senses, usually without a special suggestion of opulence in the thing enjoyed or sensuality in the pleasure.

• and learn the 【luxury】 of doing good
  —Goldsmith
• Mark decided to walk back by the road . . . instead of indulging himself in the 【luxury】 of once more rejoicing in the solitude of the green lanes
  —Mackenzie
• a dressing room with a marble bath that made cleanliness a 【luxury】 instead of one of the sternest of the virtues
  —Shaw