Using ATM, you prolong the time on the ground during the takeoff roll increasing the risk associated with contaminated runways (difficult control of the airplane, damages by ice, slush or pieces of snow blowed to the plane structures etc.) together with increased takeoff roll distance.
Compared to the ATM, Derate method is better on contaminated runways because you have a V1mcg lower than using the first one (due to the Derated max N1 limit that is used to compute the Vmcg: less thrust = less yawing moment in case of 1 Eng out); on the other hand, using ATM doesn't prohibit to increase thrust (windshear, engine failure) on the remaining engine up to the max takeoff thrust increasing control difficulties (while with Derate you can't without being "out of limitation" and getting the situation worse with a low Vmcg).
Derate method could reasonably let you take a higher maximum takeoff mass than a full thrust normal rating takeoff on contaminated runways. (the airline I work for doesn't use the Derate method, those are just some informations I recovered deep in my odd mind
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