As long as your external drive works on both machines & OSes why don’t you simply copy your desired files from the source machine’s drive and copy to the target machine’s drive? I can’t see a simpler solution for you.
If your external hard drive can be connected to both your desktop and your laptop (possible through a USB port?), then you could transfer the needed files from your desktop to the external drive, then connect the external drive to the laptop and transfer them to the laptop.
I’ve also used a USB thumb drive to transfer files. It depends on the total size of the files you need to transfer.
Ethernet is pretty fast. You can plug both into a network (router or switch) or just use a crossover type cat-5 cable between them. This assumes both have a network interface card.
RancidYakButterTeaParty, The very easiest approach without using your external hard drive as the intermediary transfer device is to simply pull the hard drive out of your PC, stick it into a USB external hard drive case, then hook it up to your laptop. Directly transfer whatever files you want. When you’re through, just put the hard drive back into your PC.