I have the same issue (same symptoms i.e. device management sees the
drive and its controller but disk management cannot see it nor can
diskpart.) although my problem arises from a seagate BUP suddenly
refusing to recognize on windows after plugging into mac. As I don't
see references to a similar question being asked, I see the need for
this question to be answered.
External USB drives are sort of a different area. Often with problem drives, the drive is detected as a mass storage device because the enclosure tells Windows it is a mass storage device, this however tells nothing about the state of the drive inside the enclosure.
Does it spin?
First diagnostic step could be: Use your ears and listen if you can actually hear the drive spin up when you connect it.
If it doesn't, you'd need to separate the drive and the enclosure and see if the drive powers up either connected using it's native interface (SATA port for example) or by using a different SATA > USB adapter. Note that for 3.5" drives you'd need one with an external power adapter.
It doesn't spin
Again the cause can be the enclosure (not providing power to the drive) or the drive itself for example due to a blown TVS diode. So again separating the drive and enclosure should allow you to pinpoint this.
A blown TVS diode can be 'fixed' often by simply removing the diode.
However the drive now runs without protection so it should be imaged/cloned ASAP and then no longer be used. For more information: http://www.users.on.net/~fzabkar/HDD/TVS_diode_FAQ.html
It spins
If you can get the drive to spin up using whatever method there should be no reason for Disk Management to not display the physical drive regardless what's on it. Disk Management may present prompts, prompting you to do something, it may display RAW partitions but it should be able to show the physical drive.
To rule out any PC specific issues, hook drive up to another PC.
If the drive spins, but does not ID at all, the drive's controller is the first suspect.
It spins
Try the drive with Linux. Apart from ruling out causes using different hardware, a different OS might make a difference. For example if Linux detects the drive, then you know it's not a drive issue, but possibly a Windows issue.
It does not take much for Windows to drop a USB drive, while Linux may still work with it
It spins, is detected, but with wrong capacity
This would tell us the controller is okay, but the firmware (which it needs to read from the system area on the platters themselves) is corrupt.
Long story short
Long story short, by using different hardware combinations you try to see if it's PC related. Use different operating systems to determine if the issue is OS related. By separating the drive from the enclosure you try determining if it's related to the drive itself or the enclosure.