It seems to me that the first (נשני) is an active forgetfulness, and the second (שכחה) is passive.
This we see from the gemora in Sanhedrin 102b which expounds that the name Menashe (ben Chizkiyohu) signifies that he caused Yisrael to forget Hashem. And even according to the Maharsha who disagrees with Rashi that נשני is an expression of forgetfulness and suggests that it is related to the expression גיד הנשה and is an expression of removal, it still indicates an active process.
In contrast we see from the mitzvah of שכחה - leaving forgotten sheaves in the field for the poor - that this type of forgetting is passive. Thus we are commanded to actively remember to erase the memory of Amalek - to erase Amalek to the extant that they will no longer be remembered - because otherwise we will in the course of time passively forget that we are commanded to do this.