Are both Computers USB 2.0 ? Older machines have USB1.1 and are too slow for data transfer on external drives but normally as you plug in a warning tells you this.
If you go to Control Panel-Click System-Click Hardware-Click Device Manager. At the bottom of the list is Universal Serial Bus Controllers.
if you have USB2 installed,you will see an 'Enhanced' USB host controller present in the Universal serial bus controllers section. If the host controller listed isn't 'Enhanced' then you have USB1
Just thought. Check your laptop as well you should see Enhanced USB Controller listed as you say this works in your external HDD.
In theory USB 2.0 devices should work on a USB 1.1 socket but in practice you will find that some do not.
In case I do not check back on this Question. If your Desktop PC is more than 3 or 4 years old it probably will not have USB 2. If you have a spare �10 and a spare PCI slot you can fit a USB2 PCI card with 4 slots. This will sort it out. If your PC is a lot older it may not have PCI slots but the larger ISA slots. No solution as far as I know to those.