Does the dishwasher drain hose connect to a sink? If so, does that sink drain properly?
If yes, then here is my initial thought: you may have a partial blockage somewhere down the line, or perhaps a misaligned pipe somewhere - bottom line is, the drain flow is being "choked" at some point. Or perhaps each of the two appliances you mentioned have their own individual partial clogs in their respective drains.
Here's how it works: the drain line is only partially blocked, so gravity-powered wastewater seeps through slowly; but it gets through before enough of it has arrived at the blockage to cause major problems. The washer pump and (to a lesser extent) the dishwasher pump, on the other hand, have two major factors: a) they're pumping a LOT of water into the line, and b) they're pumping it VERY FAST. As a result of so much water trying to get through the narrow opening in the blockage so quickly, the water builds up at the blockage very quickly, sort of like a holiday weekend traffic jam (except that I would never want to spend my holiday weekend in the place where that water is headed

). This will (obviously) result in a backup.
Before we go any further, please provide some more details: WHERE is each appliance backing out of (standpipe, floordrain, nearby fixture, etc.)? Does the water back out of the same opening for both appliances, or does each back up at its own location? Are they close to each other - i.e. would it be reasonable to assume that they're both draining very close to each other, into the same line? And by the way, what happens "after the storm" - i.e. once it does back up, does it just sit there, or does it eventually drain very slowly?
Cheers,
Moishe