Leave it on or turn it off?

Every time you turn your computer on it is going to "spin up" and the light is going to come on. Just because the light goes out and it idles so that you can't hear it anymore doesn't mean it isn't still on. It is. You can verify this by using windows explorer to view it and its files.

you could disassemble the case and remove the drive. it could then be connect via an external controller such as(http://www.tigerdirect.com/applications/SearchTools/item-details.asp?EdpNo=2477926&CatId=3770) and test if the drive itself is good in which case it is the controller in the case.
cheapest way I can think of to diagnose these external enclosures

GL
Don't forget to vote if I have help you,
TY
Jammer

my computer dosn't recognise my maxtor external drive

go to device manager/USB Root Hub/right click/properties/Power Management/uncheck the box that says "Turn of this device to save power"

Try a different usb cable.

Try connecting it to another computer.
If it is not detected, then either the disk-drive "inside" the unit is dead, or the "adapter" (USB-to-Disk-Drive) inside the unit is dead.

If the latter, disassemble the case, remove the disk-drive, and attach it as an "internal" hard-drive to your computer.

First, try another USB Cable. If that does not work properly then I would suspect the box or the drive has failed. If you just bought it, return it to the store and complain. They may replace it. As for your data?

Now if you want to become techie you can remove the hard drive from the box and attach it to you computer either copy the files back if you do not have them on your computer or format the drive if you have your files. Then put the drive back in the box and try to get a refund or buy a different external. I like a metal box with an AC cord plug. Best of luck.

type diskmgmt.msc in run then right click and assign drive letter

Try some other USB device (memory-stick, printer, camera) in the USB slot in the Vista computer, just to "prove" that Vista will recognize _any_ USB device connected to that slot.

right click my computer
select manage
select disk managment
right click on the maxtor drive then select change drive leter to "L" if you like.