IMHO Santorini is miles better than Mykonos, which had lots of "unloved" places outside the main town. Both are touristy but for some reason Santorini feels much more relaxed and less commercial than Mykonos. Mykonos is where you buy the cheap T shirts but Santorini is where you sit in a nice cliff-top restaurant and just absorb the vistas in near silence.
We stayed at Anastasis Apartments (really a boutique hotel) which was voted the world's most romantic hotel by Tripadvisor last year, and it was great - but really popular so need to book about 6 months in advance. Check out pics at anastasisapartments.com
Be aware that most of Santorini is open only 6 months of the year (from about April/May to Oct. I think)
Santorini has spectacular cliff top walks connecting the towns - you get the buzz of tourist crowds in Thira and Oia but walk a short distance and you can virtually have the sea views to yourself.