- Sefer Yereim says that these are actually two parts of a single mitzva. The obligation to remember Amalek is simply a requisite for bringing about their eventual destruction
- The Rambam Sefer Hamitzvot(מ"ע קעט) says they are two separate mitzvot. He ultimately agrees with the Yereim though that the mitzva to remember is there to help implement the mitzva to eliminate Amalek
- There was some discussion about whether the Rambam's Mishneh Torah(מלכים ה:ה) took a different approach to that in Sefer Hamitzvot. I wasn't really convinced by those who were arguing there was a difference--the Rambam is very succinct here and what he says sounds compatible with his approach there.
- The Sefer Hachinuch sounds basically the same as the Rambam, although he does mention an independent reason for the mitzva to remember: so that we should know that Hashem hates those who want to harm the nation of Israel
- The Minchat Chunuch goes further, suggesting that the two mitzvot are independent of one another
So ultimately we have two basic answers:
- The mitzva to eliminate Amalek is the dominant one, with the mitzva to remember being subservient to it
- The mitzvot are independent of one another, despite their similar context
The Rav brought 3 nafka minot of this machloket:
- The Sefer Hachinich states that women are excluded from the obligation to remember because it's the men who are actually required to make war on Amalek. The Minchat Chinuch argues that the mitzvot are independent so women should be obligated to remember Amalek(he also argues that in any case women are obligated to make war on Amalek too)
- Once Amalek ARE destroyed are we still obligated to remember them?
- Rav Frank in מקראי קודש brings the question of whether one can fulfill his obligation if he reads the parsha at the end of Beshalach instead of parshat Zachor. If the mitzvot are independent, then yes. If not then only Zachor mentions the obligation to eliminate Amalek, so only it will suffice.