Apps like FCE frequently store stuff in /Library/Application Support/ which would not get copied with a drag-and-drop. The app would still run from the new location, so long as you were booted from the drive with the additional files in its /Library folder; the test would be to boot from a fresh install and see if it launches.
Using Migration Assistant to import apps and their supporting files will get this stuff for you automatically, but if you're trying to clear out junk that might defeat the purpose of your clean install.
You could also locate and move the files manually, but frankly if you've got a legit FCE license it's easiest (and safest) to just reinstall it from the original disc.
