In this question it discusses the background of saying both Zeicher & Zecher in Devarim 25:19, 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, (b. bathra 21a), records a discussion wherein Joab 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.")
There are multiple kinds of Segolites (penultimately stressed two syllable words marked with Segol/Tzere-family vowels). Some are Segol-Segol (6-dots) and move to Tzere-Segol (5-dots) when in construct form (Neder -> Neider : Vow -> Vow-of), while some start with 5 dots and remain that way in construct from (Seifer -> Seifer : Book -> Book-of), while others start with 6 dots and remain that way in construct from (Beged -> Beged : Clothing -> Clothing of). These types develop a Chirik when they get a suffix (Nidro = his vow; Sifro = his book; Bigdo = his clothing).
While there was a small amount of discussion in medieval times which category זכר is, it turns out it's of the second type. So there is no word Zekher (6 dots) and it hence has no meaning. (Someone saying Zekher is likely just mispronouncing Zeikher.) Zeikher (5 dots) can mean either "remembrance" or "remembrance of" depending on context (in your case it's the latter).
Of historical note: there was a time when printers were printing Zekher (6 dots) in various places in Tanakh and/or the Siddur, rarely in conformance to any of the three categories above. Some homiletic explanations were offered for these discrepancies (eg. the first Lubavitcher Rebbe wrote that the 6 dot version represent love and kindness, while the 5 dot version represents fear; alternatively, R Yosef Steiner (Afikei Mayim 1:85) said the 5 dot version represents exile (Tzere like Tzar) and the 6 dot version represents redemption (Segol like Segulah)), but these suggestions aren't based in and don't reflect the traditional grammar.
R Meshullam Roth (Besoret Eliyahu 5) in explaining a passage in the Talmud (BB 21b) suggests that Zekher (6 dots) can be the construct form of Zakhar (male), like how 'Eshen (Exodus 19:18) is the construct form of 'Ashan (smoke). He relies on his read of Ibn Ezra there who claims 'Eshen is essentially an exception which happens occasionally where words borrow other forms' constructions. There are a lot of problems with this. No one ever suggested this as an explanation of that passage before. There is no reason to think Zakhar has exceptions if we've never seen one before. More importantly, 'Eshen is probably not the construct form of 'Ashan, but rather of 'Ashein (cf. Exodus 20:15, Isaiah 7:4), like Yerekh is the construct form of Yareikh (Genesis 32:33), Ketef/Kateif, Geder/Gadeir, etc. Rashbam (Exodus 19:18) notes this and says 'Ashein means "smoky/smokiness[/smog?]" (פומיאה בלע"ז fumes; while we might not appreciate it, furnaces back then were really smoky). There is no evidence of a word Zakheir (male-y (?)) nor would such a word, even in construct state, fit in the context of that passage in the Talmud. Rather, as above, Zekher is not a word.
Rabbi Soloveitchik was reported to repeat these words in Ashrei as well. This practice is recorded in this online siddur, along with the meaning of each word.
The Rav would repeat the verse, but with the first word read “זֵכֶר”, instead of “זֶכֶר”, much as we do for the same word when reading Parashas Zachor. One means “that which reminds you” and the other means “a memory”. It is unclear which means which, and although the Vilna Gaon taught specific meanings for each reading, his students’ writings conflict as to what his opinion was as well. Notably, Rav Chaim Vilozhiner quoted the Gaon as saying that “זֶכֶר” is the one that means “a memory”, which is the meaning intended here – “They will give expression to their memory of Your goodness”.