Why cannot I use the USB port directly to communicate via PC with STM32F4-DISC
In both boards, communication between the PC and the microcontroller is mediated by another microcontroller. 

On the Arduino board, communication between the target micro and the interface micro happens through the serial port of the target, while on ST boards communication happens through SWD interface.

ST used SWD interface because it allows for much more sophisticated control and debugging, while the serial port is somewhat limited on that sense.

When you are not programming the target, on Arduino boards, the serial interface is free to be used as is; the interface chip is seen on PC side as VCP (Virtual COM Port), so the PC thinks it is connected to a serial interface, and sends (and expects) serial signals, that are relayed directly to the target micro. This is not possible through SWD though.

While you do not specify the exact discovery board you have, most if not all of the target MCUs of the STM32F4 series have an onboard USB controller, that can emulate VCP with the ST software stack. If there's a second USB on your board, usually "USER USB", odds are you can use that one, but you will need to use (and understand) the software stack, at least to some extent.