What could cause an 01 Mazda 626 to not start with a fully charged battery, but will start with different batt?

I have a 2001 Mazda 626 2.0L. The car won't start, you turn the key and hear a click-suspecting the start solenoid, but the radio will turn on and everything. I threw the battery on the charger and after a few minutes it showed fully charge. I tested a different battery in the car and it started right up. I had the battery load tested and it showed fine. What could be the cause?|||Check the alternator. See below.|||Not all the batteries have the same specifications. Could be amperage or voltage differences.|||the battery that wont start the car most likely has less cold cranking amps (ccr) than the other battery. thats why it dies. not strong enuf.