A start capacitor is used for momentary starting of a motor. It is rated for a low capacitance value to deliver the required current for the short time it's on.
Run capacitors are more commonly found in HVAC systems and are designed to stay on the circuit for continuous use. This increases their efficiency and saves energy costs in the long run.
Run capacitors typically have a much lower failure rate than start capacitors. They also have a smaller capacitance value than start capacitors. In most cases, they are the better choice for air conditioners and fans. They are also less costly to install.