What is the difference between Assumed Temperature and Derate takeoff thrust?
The difference is in the minimum control speeds ($V_{mcg}$ and $V_{mca}$).

In Assumed Temperature takeoff thrust, the pilots are allowed to advance the thrust levers to full rated takeoff thrust. However, in Derated takeoff thrust, the levers should not be increased beyond the fixed decate limit, otherwise a engine failure may result in loss of directional control.

From Boeing 737 FCTM (my emphasis):


  The fixed derate is considered a limitation for takeoff. Takeoff speeds consider ground and in-air minimum control speeds (VMCG and VMCA) at the fixed derate level of thrust. Thrust levers should not be advanced beyond the fixed derate limit unless conditions are encountered during the takeoff where additional thrust is needed on both engines, such as windshear. A thrust increase, following an engine failure could result in loss of directional control.


The minimum control speeds are calculated based on the full rated takeoff thrust. The rated thrust is lowered in Derated takeoff but not in Assumed Temperature takeoff. Therefore, by calculating the minimum speeds based on a lower engine rating, the pilots can be ensured that sufficient directional control can be maintained on a slippery runway, whereas there is no such guarantee for Assumed Temperature takeoff.

On a dry and long runway, there is no operational difference between Assumed Temperature and Derated takeoff.

From the Boeing 777 FCTM (my emphasis):


  Reduced takeoff thrust (ATM) may be used for takeoff on a wet runway if
  approved takeoff performance data for a wet runway is used. However,
  reduced takeoff thrust (ATM) is not permitted for takeoff on a runway
  contaminated with standing water, slush, snow, or ice.


And:


  Derated takeoff thrust (fixed derate) may be used for takeoff on a wet
  runway and on a runway contaminated with standing water, slush, snow, or
  ice.


The Boeing 737 FCTM has a nearly identical description.

The A330/A340 FCTM explains that the slower $V_{mcg}$ / $V_1$ can also allow increased MTOW on short runways:


  When taking off from short or contaminated runways where ASDA [Accelerate-Stop Distance Available] is the limiting factor, a reduction in the minimum control speeds may generate a take-off performance benefit and a higher MTOW.