Here's the list, in alphabetical order:
Ashford-in-the-Water: A very pretty village with plenty to see...and a good base for accessing the Monsal Trail. The article states that Sheepwash Bridge is the most photographed bridge in England.
Bakewell: The unofficial capital of the Peak District; plenty of quirky shops, pubs, and cafes.
Castleton: Tourist honeypot - easy to reach from Sheffield. Plenty of shops and pubs. The village specialises in the production of blue john jewellery...the mineral is found nowhere else on Earth.
Edale: It wouldn't be on my list. It's surrounded by stunning countryside but the village is nothing special.
Edensor: A unique village on the Chatsworth estate. Every cottage is different yet they were all built at the same time.
Eyam: The 'Plague Village' A fascinating history and an uplifting example of self-sacrifice for the common good. Plenty to see and do, and some pretty, yet quite easy walks in the area.
Hartington: I've only visited the once and wasn't that impressed with the place.
Hathersage: More workmanlike than most places in the Peak District; it has nice areas though. Easy access to nearby climbing areas on the Eastern Edges.
Tideswell: Plenty of shops and cafes, and a large parish church known as the 'Cathedral of the Peak.'
Tissington: I've not been yet, so can't comment.
Personally I wouldn't include Edale and Hartington on the list and would replace them with Monyash and Froggatt.