There are a few things that keep a rocket stable that are not present on a balloon. Either one will tend to rotate or change trajectory if the thrust is not aligned with the object's center of mass. Rockets are carefully designed so that this is generally true, and some rockets have a gimbal system that allows the thrust to be redirected at carefully controlled angles. A balloon has no such control - the air just shoots out in whatever direction the neck happens to be pointing, resulting in an uncontrolled thrust vector. Air exiting the neck will affect the neck itself, so thrust vector will change direction rapidly and unpredictably.
Additionally, many rockets have fins near the bottom which helps to keep orientation. By putting the "center of drag" behind the center of mass, you have a object that will naturally tend to self-right in flight - if the object starts to rotate, increased drag on the rear of the vehicle will pull it back in a straight line. This is why you see fins on many flying objects such as rockets, arrows, or darts.
Overall, a balloon does not have the same aerodynamic stability that a rocket does, nor does it have any control over its thrust vector. A rocket is designed to be aerodynamically stable, and has a much more controlled thrust vector.