Ask21:
A flat spin is a spin where the AOA is almost 90 degrees, so you are headed virtually straight down but with very little pitch.
Recovery is as follows:
Have all of the pax crowd into the front. Then have them all return to their seats as you level off with airspeed, if they can do it under the G's.
A deep stall can occur in some in some designs, generally T-tailed aircraft, and unlike a flat spin, the aircraft is steeply nose down. In this condition the wings block airflow over the tail, so you have no pitch or yaw control.
They say the chances of recovering from a deep stall are greater than from a flat spin, but I've never flown an aircraft with deep stall characteristics, so I don't know the recovery procedure for that one. I imagine it varies by aircraft.