As I thought, you're saving the playlist. But if you only select the song & press Save As ... you're saving an empty playlist file. The save as... only applies to the playlist function. To get songs over to the playlist you need to enqueue them 1st. And remember this is only a playlist. (More about how to use LW's playlist at the bottom of this thread: How to transfer music to my iPod (& iTunes))

So you are transfering empty m3u files or just moving the playlist files but not the actual song files. What you need to do is move the actual songs to the other computer. Whilst within the LW Library window, select Saved Files & press the Explore button & the folder will open for you. Then you can copy & paste these mp3 files over to your other computer.