I've tried connecting a 2.5" external drive to an old laptop which has only USB 1. The LED on the disk lights up, but the disk doesn't seem to spin up. Since the same disk works fine on a newer laptop my guess is that the old one doesn't output enough power on the USB port. Besides looking for an external drive with its own PSU, what would you suggest? Will one of those USB cables with two connectors work? What about a powered USB hub?
As far as I know, USB 1 and USB 2 may have different speeds, but the power should be 5v on each.
If it works fine on other laptops, it could be that your laptop is outputting the incorrect power. (Or, the ports are un-powered :S... remember something similar a few years ago, but cannot find reference).
I would not risk using a dual USB cable or you could do damage.
If you have a powered USB hub, it is worth a try, but make sure it works fine on your other computer first.
If all else fails, you can look in to putting it on another machine and using Windows networking or another file transfer method.
I would have to agree that you're drive is probably drawing too much current for the USB port. When this happens, there's usually an internal current limit switch on the hardware which trips, preventing your USB device from sucking any more power (so there's no damage to the motherboard). Your drive's LED is probably green because current has been limitted, but is still on.
If you need to connect your 2.5" drive to the laptop via USB, I would recommend purchasing a cheap 2.5" PATA drive to USB converter that comes with a power adapter. Here's one on Newegg for $19.99 (USD). It comes with power adapters for 3.5" and 2.5" PATA drives, and a power brick.