There are several possible reasons for this.
The most common is a short or failure in the Lacie enclosure caused by mixing up power/AC adapters. If your two Lacie drives are different models and you accidentally used the wrong adapter you may have damaged the circuitry.
The second possiblilty is that your power adapter or USB/Firewire cables could be damaged. Try switching USB/Firewire Cables First. If this fails to work then check if you have a smae sized adapter with the exact same Output Voltage and Amps on both adapters. Only if they are identical try switching adpaters from the working unit. If they are not proceed to the below solution.
Finally if the unit was dropped, bumped or unplugged while it was running it is quite possible that you may have damaged either the drive or the file structure on it, either of which would cause it to stop showing up.
Try the cable swaps if able, if they work, replace the USB cable or replace the entire enclosure if the power adapter is the issue.
If neither works then you have either a failed hard drive or failed enclosure. Go to your local PC store (not a big box and yes PC, they tend to have more experience and parts to work with in this type of situation), have the techs there remove the drive from the Lacie enclosure and place it in a new enclosure for testing. They may not be able to see your files but they will be able to tell if the drive is working. If it is, buy the enclosure, then connect the drive back to your Mac. If it works you're done.
If the drive works but the Mac still doesn't recognize it(doesn't show up on the screen), then comment back and I will get you the link to a program you can use to retrieve the lost files
If the drive simply fails to show up at the store in the new enclosure you will need the help of a pro to recover any important files. If that is the case reply with your location, Country, State and City and I will recomend an appropriate facility for you.
Hope this helps FixYa,