In Zeicher vs Zecher it discusses the reasons we say both, however I was wondering what is the difference in translation between Zeicher & Zecher?
The standard understanding is that it means "memory of" either way, just a question of how it's vowelized. (You'll find both vowelizations in different texts of Psalm 145, known to most of us as the daily "Ashrei", on the line "zecher rav tuvcha...").
But just for fun, the Gemara records a discussion wherein King Shaul thought the commandment only applied to the male Amalekites. The standard reading of that Gemara is he read "zachar" (male); but it's been suggested that "zecher" could be misinterpreted as a possessive form of "zachar" (just as "eshen hakivshan" means "smoke of the kiln", "eshen" is the possessive form of "ashan.")