Arduino Micro vs. Pro Micro
I actually think you are meaning what is the difference between the Arduino Mini and the Arduino Pro Mini, since there is no such board as the Arduino Pro Micro. However there is a board called the Arduino Micro and also the Sparkfun Pro Micro, so this answer will look at all four.

Arduino Mini
  

Arduino Pro Mini


Arduino Micro
  

(Sparkfun) Pro Micro
  

Microcontroller
Arduino Mini: ATmega328
Arduino Pro Mini: ATmega328 (ATmega168 on older boards)
Arduino Micro: ATmega32U4
Pro Micro: ATmega32U4  

Memory
All four have 32KB of Flash and 1KB of EEPROM
Both Minis have 2KB of SRAM where as both Micro boards have 2.5KB  

Digital I/O
Both Minis have 14 (of which 6 provide PWM output)
The Arduino Micro has 14 also (of which 7 provide PWM output)
Some of the analog pins can also be used as digital I/O
The Pro Micro has 12 (5 PWM pins)

Analog Inputs
Arduino Mini/Pro Mini: 8 (some boards only break out 6 analog inputs)
Arduino Micro: 12 (6 dedicated and 6 from digital I/O pins)
Pro Micro: 9 channels of 10-bit ADC

USB
Arduino Mini/Pro Mini: Requires FTDI cable or Sparkfun breakout board to provide USB power and communication to the board.
Arduino Micro / Pro Micro: Native USB  

Clock
All 16 MHz
(Some 3V3 variants run at 8MHz)

Power
Arduino Mini: 7-9 V
Arduino Pro Mini: There are two version of the Pro Mini. One runs at 3.3V and 8 MHz, the other at 5V and 16 MHz. VRAW  can accept voltage up to 12VDC.
Arduino Micro: USB or external supply of 6 to 20 volts (recommended range is 7 to 12 volts)
Pro Micro: can accept voltage up to 12VDC

Interrupts
Arduino Pro Mini: The Atmega328 has 2 hardware interrupts available
Arduino Micro / Pro Micro: The ATMega32u4 has 5 hardware interrupts available
credit: Enric Blanco