The "show inventory" command displays the chassis serial number. That is the one used when registering the device for Cisco support (Smartnet) and reflects what's printed on the sticker affixed at the factory. If you have support and need to open a TAC case, this is the number you should reference.
The "show serial-number" command is the motherboard serial number. It is equivalent to the one shown on an ASA when you use the "show version" command. In fact if you switch to the LINA prompt ("system support diagnostic-cli") you can run "show version" there and also see that serial number.
This latter number is not really used much (at all?) when you are running FTD. If you were running an ASA image and installing a PAK-based license (i.e. AnyConnect) you would use that number during license registration.
