The word "frank" (or its cognates) is an interesting one. All over the world, it refers to Westerners (as viewed by locals), but in Israeli slang, it's a derogatory term for Sephardic / Mizrahi Jews (as used by Ashkenazim). How did this happen?
According to this Philologos post, and this Language Hat post, the French were the ones leading the initial Crusades, and so they became known as the standard European foreigner. Philologos mentions the following cognates in many languages - all over the world:
Greek frangos, “Westerner”; Turkish frenk, “European” (frengi in Turkish means syphilis, for which the Turks had Europe to thank); Syriac frang, “European”; Persian ferang, ditto; Amharic frenj, “White Man”; southern Indian farangi or pirangi, “European” or “White Man”; Thai farang, ditto; Cambodian barang, ditto; Vietnamese pha-lang-xa, ditto; Malaysian ferringi, ditto; Indonesian barang, goods sold by a foreign trader; Samoan papalangi, “foreigner.” (Other derivations for papalangi, however, also have been given.)
In fact, the name might even extend beyond our planet. He mentions the Ferengi of Star Trek, whose name might have the same source. (We've seen Star Trek here before).
In Hebrew slang, the term franji פרנג'י means "to dress fancily, in a European style". But this phrase is not in common use today (in fact, I'm not sure if I've ever heard it myself). However, the pejorative frank (actually better spelled frenk), which sometimes in Hebrew is still spelled in the Yiddish style פרענק instead of the Hebrew פרנק, is still heard (if not in polite company).
Why in this case are the Ashkenazi westerners calling the "local" Sephardim by this term? Ruvik Rosenthal writes here that the usage derives from the Spanish word "Francos", which had the same meaning we've seen before - Western Europeans as viewed by people in the East. In this case it referred to Sephardic Jews who migrated to the land of Israel from Spain and the Balkans. The local Jews referred to them as Europeans, and when the Ashkenazi Jews immigrated to Israel, they referred to all Sephardim as "frenks" - and the sense became much more insulting.
A different form of the root פרנק, which is much more positive, but unrelated to the modern use is found in the midrash. For example in Midrash Tanaim on Devarim 32:2 it says that the words of Torah are מעדנים and מפרנקים - "refreshing" and "pampering". However this root is simply an expansion of the root פנק - also meaning "to spoil, pamper", and the Midrash Sifrei on the same verse uses מפנקים instead of מפרנקים. The root פנק appears once in the Bible, in Mishlei 29:21 - מְפַנֵּק מִנֹּעַר עַבְדּוֹ - "a slave pampered from youth". In Modern Hebrew we see the word pinuk פינוק with both the positive connotation of "pampering" and the negative connotation of "spoiling". Like with the previous meaning, what can be fancy and pleasant to some, can be overindulgent and arrogant to others...