Playlists are odd things because they are simply a list that points the player to where to find the songs. So the player knows exactly which directory to find the songs. If you move a playlist to a new computer, then I guess it depends on how intelligent your player is in reading playlists or whether it reads them at absolute address ... meaning the identical directory.

My suggestion would be to try to set it up so your old audio files are in the identical locations to the original comp.