Circuits in audio amplifiers to protect speakers
These circuits are often called "soft-start" or "pop-suppression" or "speaker-protection".  There are probably other names but I don't think there's a standard.

A popular protection circuit uses relays to connect the amplifier output to the speaker connector.  There are several ways to control the relay:


Keep the relay disconnected for x seconds after power-on, giving the amplifier time to settle, then close the relay to connect the speaker.  This alone would eliminate your pop at power-on.  You can also disconnect the speaker when the power is turned off (but before the amplifier power supply drops too low to work) to eliminate a thump at turn-off.
A more sophisticated relay-based protection circuit would low-pass filter the output signal to look for any DC component and disconnect the speaker if it detects it.  DC can burn out any non-AC-coupled speaker (like a woofer), and audio amps don't need to go below ~20Hz, so any DC component would indicate a fault condition and the relay would open to disconnect/protect the speaker.  Typically DC faults are caused by failed transistors or the collapse of one of the supply rails.  Depending on the design, this could be used WITH the above delayed-solution or instead of it (since the thump during power-on is usually a low-frequency transient that the DC detector could catch).
Many amplifier designs also monitor the temperature of the output transistors and disconnect the speakers if they go above the max temp the design allows.


A lot of audiophiles believe relays color the sound, and the resistance of relay contacts can degrade over time, so some amp designers add solid-state circuits that remove the biasing from the output transistors to effectively disconnect the amp, and provide a soft-start function during power-on.  This might not provide quite as robust protection as a relay, but it's a good solution.  

If your goal is to fix the amp with the turn-off thump, there are two options:


Add a relay and a circuit that detects when the power supply starts to fall that disconnects the speaker.
Since one of your amps doesn't have the problem, try to find out what's different between the amps.  If you can find and post a schematic I could tell you what to probe for - we might be able to troubleshoot it with just a voltmeter...  It's possible that the noisy amp has a protection circuit that's not working correctly, but it's also possible that the quiet amp just happens to be more "balanced" during power-down (the supplies come down more symmetrically or in a way that turns off the correct output transistor earlier)...