Balloon framing is the earliest form of light wood framing. This framing system uses closely spaced wood studs, joists, and rafters. Wood studs run continuously for the full length of two stories, from the foundation to the roof. The two major cons of balloon framing are that long studs are less efficient to erect then shorter ones used in platform framing and the continuous spaces between studs act as chimneys during a fire, unless closed off with a fire-stop. However, balloon framing does offer some advantages as well. It can be useful when creating two story spaces and you are less likely to have cracking develop in your interior and exterior finishes because of the minimal amount of vertical shrinking that occurs with this method.
Platform framing developed out of balloon framing. This framing system starts with a floor platform on top of which stud walls are built. A second platform is built on top of those load bearing walls and additional walls are built on top of the second platform. Those walls support the roof. The pros of this system are that it uses shorter pieces of lumber than balloon framing making it easier and cheaper to construct and the platforms between each floor act as a fire-stop. The major disadvantage of the system is the shrinkage that can occur vertically in the platform causing damage in the interior and exterior finishes.
Platform framing is going to be more cost effective and easier to construct. However, because of the minimal amount of vertical shrinkage with balloon framing you might use this method in cases where you are worried about cracking in finishes like stucco for example. Overall, I think in most cases platform framing is likely to be more desirable than balloon framing.
Fundamentals of Building Construction by Edward Allen and Joseph Iano and Building Construction Illustrated by Francis D. K. Ching are two sources that I found helpful for this topic and are great places to look for more information.