You need the 360 wireless receiver to use your wireless 360 controller with PC, cable or not.
Play and charge cable is only used to transmit power, not data. It works with Xbox because Xbox has the receiver built-in.
To quote Microsoft support article on this issue:
A standard Xbox 360 wireless controller can be connected to a Windows PC, but you must have an Xbox 360 Wireless Gaming Receiver to play games, even if you have an Xbox 360 Play & Charge Kit.