Sweet potatoes are a traditional staple, and much better for you than regular white potatoes, but as there are no contests for the "most nutritious" vegetable, I can't verify the claim you heard. Nor should you believe that you could live on sweet potatoes alone - you wouldn't do well on such a limited diet. (You wouldn't get enough protein, nor would they provide essential fats.)
However, sweet potatoes are packed with nutrients, including fiber, copper and carotenoids, precursors of vitamin A found in fruits and vegetables. Sweet potatoes also rank much lower on the glycemic index scale than white potatoes, which means that even carbohydrate-sensitive individuals can include them in their diets.
You may be interested to know that vegetables labeled "yams" are really varieties of sweet potatoes, and all are cousins of morning-glories that come in different colors and textures. Moist, orange-fleshed varieties are often called "yams" while drier, yellow-fleshed ones usually are called "sweet potatoes." Both types are unrelated to regular white potatoes. (True yams are gigantic tropical tubers seldom seen in U.S. grocery stores.)
Sweet potatoes are so good for you that I encourage you to eat them often, at any time of the year. They're not just for holiday meals where they're often served loaded with added sugar and fat. You're better off baking moist, orange-fleshed sweet potatoes in a medium oven (about 350 degrees). They should need no added sugar. You can cook drier types in the microwave (five to 10 minutes on high, depending on size; be sure to prick them first with a fork). You can also pan fry sweet potatoes in a little olive oil. Or you can roast them: just peel and cube them, toss in olive oil, salt, and pepper and spread in a baking pan (do not crowd). Roast them in a 450 degree oven until they are nicely browned (stir them every 10 minutes to make sure all sides are browned). Try these recipes: Roasted Root Vegetables and Vegetarian Shepherd's Pie.
Andrew Weil, M.D.