Power = Current x Voltage
The voltage of the batteries plays a huge difference in whether electronics function properly.
Each of them is built to run using a particular voltage - 3.3V, 5V, 6V, 7.2V, 11.1V, 12V, 19V and 24V are some common operating voltages for electronics. Some devices have a voltage regulator that shift input voltages to a safe level. This allows you to use a higher voltage power supply to power your robot/computer/sensor. For example, the Arduino Uno has a voltage regulator that accepts voltages between 7-30V and converts them to the 5V that the microprocessor runs at.
From a motor standpoint, decreased voltage means lower power output, which means that either your motors won't turn as fast or they won't be able to output as much torque.
The best thing for you to do right now is look up the datasheets online for your quadcopter or the individual components and find out what voltage each of the parts run at. If your second quadcopter needs anything higher than 7.2V, then there's your problem. If you have trouble deciphering the datasheets, you're welcome to edit your post to include the links and we'll try to help you understand them.