Well from my understanding, automatic climate control always turns the air-con on and off as required unless manually overridden. The air con indicator on the Alfa doesn't change, it's always red when in auto mode.
Full-auto mode on the early 159s means that all functions are being controlled automatically, including fan speed, air con and air direction. Manual setting of these functions, e.g. changing the fan speed will drop it back to "Auto" mode. The air con and air direction are still being managed automatically but you have decided to take control of the fan. You will only get rid of the "Auto" display if you override all three functions, fan speed, direction and air con.
As I understand it, later 159s substitute "Full Auto" for "Auto" and "Auto" for no display.
NB. The system will still increase the fan speed proportionatly if it detects a change in conditions, e.g. misting up or increased sunload, internal or external temperature. For example, full-auto sets fan to 3. I override it by setting it to 2. Sunload increases so fan is increased by 1 regardless of my previous setting. Sunload decreases and it gets reduced by 1.
I could be wrong however, I'm only going by past experience of climate control systems.
PS. Air con is usually turned off at low temperatures (say below 3 degrees) to prevent "compressor icing". The Automatic Recirculation function doesn't appear to work below about 6 degrees, presumably to avoid misting up.
PPS. The auto mode usually leaves the air con running. The only reasons it turns it off is due to low external temperature or for maximum power if it detects full throttle (depends on the car).