The M16A2 (AR-15A2) is selective fire. (SAFE, SEMI, BURST).
The M16A3 (AR-15A2) is full auto and was issued to Special Operations personnel who needed the ability to lay down supressive fire (SAFE, SEMI, AUTO). It was used by the US Army Rangers, US Special Forces, Navy SEALs, etc.
The M-16A4 (AR-15A3) is selective fire and has the C8A1/M4 style "flat top" Picatinny Rail and the SOPMOD M5 Rail Interface System forend(like the SFW and SFSW). It is used by Mechanised Infantry units and may serve with Special Operations forces when they are in wide open areas like the desert.
The US Army‘s TRADOC website has a manual for the M-16 series that gives the stats for the M16A1, M16A2/A3 and M16A4.
Since most people have never seen the M-16A3, they don‘t know about its existence. Civilians also get confused by the designation for the civilian AR-15A3 (the "flat tops") and think the M16A4 is the M16A3.