Depends upon what kind of appliance you're talking about. Washer/Dryer? Fridge? Dishwasher? Stove? Those are just the obvious ones, both manufacturers make even more appliances than those.
My mom swears by Maytag, but you'll pay more than Whirlpool. All of my major appliances are Whirlpool and they've all worked flawlessly... Individual models of both brands are highly rated by CR, it depends upon the models you're looking at.
FWIW Whirlpool on average is one of the top two or three brands in terms of "frequency of repair" for major appliances, frequently best of all, and in most cases better than Maytag (source again is CR).
Have you considered the Sears house brand Kenmore? I think most Kenmore stuff is made by Whirlpool, which broadens your choices and sometimes Sears offers killer deals in their weekend supplements.