Hi lwildernorva
I have stayed at both. The WA basic room is nice, plush with good finishes, however still and old building. It is not that quiet, creaky floors. If you have room facing castle you get noise from Lothian Road, a very busy road. Rooms on other side are quieter but no view.
I was given drinks voucher for 2 drinks beer or wine (diamond). No free breakfast. It is expensive and is so so. You do get spa at WA.
Doubletree, used to be Point Hotel. Remember spending many nights drinking in the local pub attached to the hotel

Nicely decorated, it used to be a department store back in the day. I was on room top floor (deluxe) big room, bit strange with long corridor opening to room, but that is good as keeps noise from corridor away, very dark rooms which I was not sure of. Bathrooms fine. Bed good and nice view to castle. Top floor rooms have smaller attic style windows, those on lowers floor have large windows. If weekend this area is noisy as lots of pubs etc in area. But good base for sightseeing.
Great breakfast incl. Large gym in basement which was very quiet. Overall I would not waste money/points on WA, it is nice as a bit of luxury but the DT does the job.