Is the new cable USB 2.0? Not all cables are equal. There are some cables with the correct connectors that do not even have the data wires passed through. they are described as charging cables.

It may sound foolish to ask, but is the power supply connected to the drive, plugged into power and switched on? You may not have disconnected it, but someone else or the cat may have.
Is there a switch on the drive casing?

Can you check the power supply and see if it is supplying 12 volts?

If none of this works, you could open up the case and remove the drive.
You could obtain a external drive case, and try it in that. That would give you a new cable and power supply so if the drive is ok, you should be able to access it.
It will almost certainly be a sata drive.