I asked about this recently. If you download Realplayer you can save youtube clips to your hard drive. Then you can move them to your flash drive. There may be a quicker way but this does work.
In the Windows folder there is a folder called Temporary Internet Files. This is the cache that lets you replay the clip without downloading it again.
Just sort by date just after you download it. Or search by size (largest first) and look for likely candidates toward the top. They usually have meaningless names but just try to run each one until you find the one you want.