Seagate harddrive 500 GB only spinning sound but not recognized by USB

First thing to try is on another computer, there after open up the external casing and take the HDD out so it can be read directly from the sata ports thereafter if it still is not working consider data retrieval options.

It sounds like the USB interface circuitry on the external drive is no good. As Pilk stated above you may have to remove the drive from the case and plug it in directly to your motherboard to recover the data. If you are unsure on how to do this or squeamish about opening up your case then you can get a USB HDD adapter like this: http://www.newegg.com/Product/Product.aspx?Item=N82E168...
It will aid you in your data recovery. I use a similar type of interface for data recovery and other work on drives. As long as there is no clicking or grinding noises coming from your external drive it is probably recoverable.

+1^ the drive disks are spinning up which means its a circuitry issue could also be the main controller board on the drive however and if thats the case you might have to buy an identical drive and switch change the boards to get the drive to function again.

Like you all said...once you have taken out the drive and it does not "work" inside the pc or another pc as well...the data retrieval options will be you last chance after the you can say goodbye to the drive...but pretty sure you can still get the info back but it most probably will cost quite a bit...soz bro