Stayed in Alila, and visited the Anantara for lunch on our way down. At the risk of contradicting earlier posters, the two are not even close, despite comparable rates.
Alila is a proper 5* which blends in with the mountain surroundings in a perfect isolated location, small enough to be peaceful and relaxing. You go there and realise the whole place just works.
Anantara on the other hand is an oversized concrete complex located off a questionable looking village, with the distinct feel of a package tour resort. The lunch was also bad, both in terms of food and service.