For my time and money, I would pick either Azores or Madeira. There is more then enough at either destination for 1 week. I have been to both, but Azores is my specialty! Even with these two destinations, there are many differences. I suggest you look over my earlier posts regarding both/each of them.
Madeira is very touristy and cost more, whereas Azores, specifically the islands I know well Sao Miguel and Santa Maria, are less so! I feel (Azores) they are very authentic to their history and culture. Also, I love seeing animals all over the Azores islands, and you will see very few animals in Madeira. I learned that Madeira is too steep to let animals graze, the Azorean islands are mainly rolling hills which allow grazing. Madeira has great walks/hikes, and the Azores as well. Beaches...hands down the Azores! Quaint villages...Azores. Ability to mingle with the locals Rick Steves style...Azores. Easier on the wallet...Azores! Amazing natural landscapes...both, but I feel more so for Azores. Easier driving around the islands...Azores. Madeira has some incredibly steep mountains and windy roads, lesser so in Azores, which makes driving a bit hairy and scary.
I hope this helps you make a decision. There are many more insights I could offer regarding either Azores and Madeira, but I think this is a good start. If you have more questions, let me know. I would youtube the destinations and check out the websites I recommend in my earlier postings to see what interests you...then go from there.
Late August in Santa Maria has the Mare de Augusto if you are into music: http://santamariaazores.net/web/category/associacoes/mare_agosto/#.VO4LiIY8KnN. There are also Ralley races there as well and local feasts in August.