Why my xbox 360 controller works only with a cable?

Youd have to have an understanding of circuits and soldering, and thats still no guarantee. Easily definately not

In addition to the instructions from XBF Chris J, make sure that you really have a wired Xbox 360 controller, and not a wireless Xbox 360 controller that you connect with a Play & Charge kit cable.

A real Xbox 360 controller that works with PC has a fixed, non-removable cable. If you can unplug the cable from the front of the controller, that cable ONLY transmits power to charge the controller.

No controller data can be transmitted via that charging cable, and therefore the PC can not detect it. But when the controller detects power coming through that cable, it vibrates shortly. Maybe it's that why the controller "only" vibrates, but is not detected
by the PC.

With the Xbox 360 controller, it only sent data wirelessly. The charging cable was only used to charge rechargeable batteries on the controller, and does not send data over the wire for the controller. You would need the Xbox 360 wireless controller
adapter (usb dongle) to connect. If the 360 controller was wired natively then that would work, but the default controllers only worked wirelessly.

Hi,

The Xbox 360 wireless controllers can't be turned into wired ones with a Play & Charge. This is because even when wired, 360 wireless controllers remain wireless. The cable only provides power, no data is sent.

This is also why they don't appear in game controllers but the generic controller does.

To use a wireless 360 controller on Windows requires a gaming receiver. And the receiver must be for 360 controllers, the official receiver for Xbox One/Series controllers won't work.

Not all 360 controllers will work. It must be the one for windows.

No that's not correct. All Xbox 360 controllers will work with Windows. The only difference with the Xbox 360 controller for Windows is that these were largely sold as wireless controllers and came bundled with the Xbox Gaming Receiver for Windows.

Unlike the Xbox One/Series X|S controllers, you can't make a wireless Xbox 360 controller wired by connecting a Play & Charge cable to it. A wireless controller connected with a Play & Charge cable are always wireless. The cable does not send data it only recharges the battery or powers the controller if no battery is present.

If you want to use a wireless Xbox 360 controller with a PC, you'll need a 360 Gaming Receiver. And, it must be the Xbox 360 receiver, the Xbox One wireless adaptors don't work.