You've got 2 very different options, there ! Depends on what you want to see. Even if Brittany has great inland places, highlights are more liked to the shore and harbors (imho...I'm not a specialist of this region). Great scenic villages, milder temperature compared to other places in France at that time (rain is an option).
Burgundy does not shut down in August !!! Plenty of things to do/see, depends on what you like. Regional focus is mainly on culture / history...food and wine. Nice climbing and mountain biking spots if you want to exercice a bit...Dijon is a nice city with a great historic center, worth a 1 day visit. So is Beaune, don't miss the Hospices and the cathedral. Easy connection by train. North from Dijon (not exhaustive): Semur en Auxois (nice medieval city), Fontenay cistercian abbey (1h drive from Dijon, worth the visit), Flavigny medieval village, Chateauneuf (great fortified medieval city on top of a hill with castle and all, very close to Dijon). South Burgundy: tons of great places if you're interested in Roman architecture, with some of the most stunning example of roman churches/abbeys, from Cluny and Paray le Monial for the biggest and most famous places in the area to clusters of villages with great churches (e.g. the Brionais area close to Paray le Monial), Solutré prehistoric site...
And, from North to south, great wine and food everywhere (these places do not shut down !). My advice would be to rent a car and drive around, get lost in the vineyards and in the small villages.
Enough to spend 5 great days (don't drink and drive though...)
