All the differences between Arduinos: Pro Mini & Pro Micro
It is more the differences between ATMega328P + FTDI (Pro Mini) vs ATMega32u4 (Pro Micro)

Apart from the minor pin differences, the peripherals for both MCUs are the same. The major difference is the ATMega32u4 has built in full-speed USB. This allows the board to function as a USB device by implementing the appropriate USB stack. E.g. it can act as a


USB to serial converter (CDC device)
Keyboard or mouse (HID device)
USB hard drive (MSC device)
MIDI controller
and more...


The LUFA library from AVR devices can be used for this but means going outside of the Arduino IDE and compiling proper C code using the latest avr-gcc tools. 

The Arduino core library has built-in support for the HID and CDC device profiles. The CDC profile is used for comms with the Leonardo, Micro, etc. while the HID profile is used for keyboard / mouse emulation.

The Leonardo / Micro boot loader implements a CDC profile for communications with the PC. This CDC profile uses Arduino vendor and product ids so unlike FTDI devices, requires an extra step to install an *.inf file on Windows. 

You would have to recompile the Pro Mini bootloader for the Micro, but even then it would require an FTDI board as programming would need to be done via the TX and RX UART pins. If you we going to go to all that trouble you could just recompile the proper boot loader to have a shorter time out. Why is difference of 1 or 2 seconds in boot loader time important?

The ATMega32u4 uses more power than the ATMega328P.