There are a few things that can try to resolve the problem.
Resetting the Dualshock 4
As mentioned in the Dualshock 4 Troubleshoot, try resetting your gamepad by pushing the reset button. It is located in a small hole on the back, near the L2 Button. You will need something thin to access it, e.g. a paperclip. Hold it for about 5 seconds, before releasing it.
You said, you have another controller that works, so you might also want to try to pair the gamepad in question using the settings menu.
On your PS4, go to Settings > Devices > Bluetooth Devices. If your Dualshock 4 in question is appearing in that list, select it, press Options and do 'Forget Device'. Now connect the controller again with the USB-cable and press the PS button to pair it.
As mlo55 pointed out, you might need to use the original cable. If you don't have it, try a few different ones. However, if the other Dualshock 4 connects over your cable, it shouldn't be problem.
Edit: There are USB cables that only carry the power lines for charging and are missing the lines for data connection which are needed to pair your controller. Hence to possible problem with the cable.
Thanks to Flater for pointing that out correctly in the comments. (I can't answer to your comment because I don't have enough reputation for that)
Maybe there is a problem with the USB cable, so try to pair your gamepad without the USB cable as mentioned in this blog.
On your PS4, go to Settings > Devices > Bluetooth Devices. On the unpaired Dualshock 4, press and hold the Share button and the PS button at the same time. Your lightbar should start flashing quickly - this means that the controller is in pairing mode. It should appear in the device list you just opened on your PS4. Select it to complete the pairing.
You might need to reopen the Bluetooth Device screen to refresh the list.