What is the difference between the various automated airport weather observing systems?
ASOS is basically a replacement for AWOS with more reporting parameters, and the first units were installed in 1991.  According to the Automated airport weather station Wikipedia page, each different type has the following parameters:

AWOS  


  
  AWOS A: barometric pressure and altimeter setting (in inches of    Mercury). 
  AWOS I: wind speed and wind gusts (in knots), wind    direction (from which the wind is blowing) and variable wind    direction (in
  degrees of the compass), temperature and dew point (in    degrees
  Celsius), altimeter setting and density altitude. 
  AWOS II:    all AWOS I parameters, plus visibility and variable visibility (in    miles). 
  AWOS III: all AWOS II parameters, plus sky condition (in    oktas), cloud ceiling height (in feet), and liquid precipitation
  accumulation (in inches). 
  AWOS III P: all AWOS III parameters, plus    precipitation type (rain, snow and sometimes drizzle) identification.
  AWOS III T: all AWOS III parameters, plus thunderstorm detection (via    a cloud-to-ground lightning detector). 
  AWOS III P/T: all AWOS III    parameters, plus precipitation type identification and thunderstorm    detection. 
  AWOS IV Z: all AWOS III P/T parameters, plus freezing rain    detection via a freezing rain sensor (Note: this configuration used
  to be called AWOS III PTZ). 
  AWOS IV R: all AWOS III P/T parameters,    plus runway surface condition. 
  AWOS IV Z/R: all AWOS III P/T    parameters, plus freezing rain detection and runway surface    condition.
  


ASOS  


  They generally report all the parameters of the AWOS-III, while also
  having the additional capabilities of reporting temperature and dew
  point in degrees Fahrenheit, present weather, icing, lightning, sea
  level pressure and precipitation accumulation.