The citric acid won't ferment the milk, if I understand how we use the language. The fermenting of milk is accomplished by adding cultures (bacteria) to the milk that will consume the sugar in the milk, converting it to acid. By adding citric acid, you are by-passing the fermenting part of cheese making. If you were to add citric acid and then let it sit out all night, whatever bacteria is already in the milk will have the chance to grow. It will continue to produce acid, but might make some pretty nasty flavors, too.
I'm not an expert, so there might be cases where you both and acid as well as adding culture to convert the sugars into acid. If that were the case, however, I would presume you'd add specific cultures, and not just let whatever happened to be in the milk do its thing.
I would guess that what you have heard about mozzarella being better fermented is talking about mozzarella made with culture as the acid producing agent instead of adding citric acid directly.
With all this said, I could be wrong and if so, someone who knows better than I do will likely fill both of us in.