You can eat latkes either with applesauce or sour cream. For that matter, you can eat them with both, but there IS a reason for the traditional options and it has to do with the rules of keeping Kosher.
Observant Jews (not my family) do not eat milk and meat products at the same meal. In fact, they tend to have two sets of dishes, silverware, etc, one for milchik (dairy) and one for fleyshig (meat) meals. There are also foods that are parve, which means they can be eaten with either kind of meal. I am not an expert on the rules of kashrut-- but you can get them here or elsewhere on the web.
The point is, that latkes themselves are parve and could be eaten as part of either a dairy meal or a meat meal. But if you were serving a dairy meal, you'd have sour cream with them. If you were eating them with a meat meal--perhaps with pot roast, or brisket, you'd have applesauce.
At my house it was always applesauce, just as it was at my grandmother's house. Though form what I recall many years later, the only meat product we had at those meals was canned salmon, and fish is considered parve. Perhaps in the days when there were many at her Hanukkah table Grandma Esther did serve them as part of a fleyshig dinner.
My mother has a horror of sour cream for its fat/cholesterol and would never have served it. My daughters didn't grow up eating it and have no taste for it. I happen to adore it, and when I make latkes at a party offer both choices.
Either way, they are delicious!