Language of Distinction

I. Double Language

R. Yaakov Kamenetsky (Emes Le-Ya'akov on Lev. 10:10) points out that distinctions in the Torah use the word "bein" (between) twice, as in "u-le-havdil bein ha-tamei u-vein ha-tahor -- and to disntiguish between the impure and between the pure." This is, he claims, a function of Biblical Hebrew.
Mishnaic Hebrew, however, uses only one "bein."

II. Havdalah

With this, he explains in two ways an ongoing debate between R. Yosef Karo (Shulchan Arukh) and R. Moshe Isserles (Rema). R. Karo writes in Shulchan Arukh (Orach Chaim 299:10) that one should recite Havdalah, the blessing at the end of Shabbos and Yom Tov, with the words "ha-mavdil bein ha-kodesh u-vein ha-chol" -- in other words, using "bein" twice. The Rema adds that women in his time frequently recited a blessing on their own, prior to the official Havdalah on a cup of wine, of "ha-mavdil bein kodesh le-chol" -- with only one "bein."

One way to explain this disagreement is that R. Karo believes blessings should be recited in Biblical Hebrew while Rema believes they should be in Rabbinic Hebrew. Alternatively, both believe that blessings on biblical obligations should be in Biblical Hebrew and those on rabbinic obligations should be in Rabbinic Hebrew. In this case, Rema was concerned for the view that women are only obligated in Havdalah on a rabbinic level and therefore proposes a blessing in Rabbinic Hebrew.

III. Challah and Shemini Atzeres

Similarly, R. Karo (Shulchan Arukh, Yoreh De'ah 228:1) writes that the blessing on taking challah from bread is "le-hafrish terumah" and the Rema states that it is "le-hafrish challah." The obligation to take a challah portion from bread is called in the Bible "terumah" but in Rabbinic Hebrew "challah." Either R. Karo believes that this is included in all blessings that must be in Biblical Hebrew or he holds that the obligation in the land of Israel, where he lived, is on a biblical level. While Rema either held that all blessings should be in Rabbinic Hebrew or that challah in the Diaspora is a rabbinic obligation.

R. Kamenetsky (on Lev. 23:36) explained similarly the debate between R. Karo and Rema (Shulchan Arukh, Orach Chaim 668:1) over the proper way to refer to the holiday of Shemini Atzeres in our prayers and blessings. Acording to R. Karo -- "Shemini Chag Ha-Atzeres", according to Rema -- "Shemini Atzeres Ha-Chag." The Torah calls the holiday Atzeres while the Rabbis call it Shemini Atzeres. Therefore, R. Karo insists on using the biblical term either because that is what we do in all blessings or because the day is biblically ordained. Rema uses Rabbinic Hebrew either because that is what we do in all blessings or because in the Diaspora we have a second day of Shemini Atzeres that is rabbinically ordained (and we use the same title on the first day to avoid confusion/denigration of the day).

IV. Exceptions?

All of that is R. Kamenetsky's analysis. There are, however, problems with it. While Gen. 1:4 speaks of differentiating between the light and between the darkness (double usage of "bein"), Gen. 1:6 refers to the firmament as being a divider between the water and the water -- only one usage of the word "bein," although the very next verse has it doubled. R. Kamenetsky (Emes Le-Ya'akov on Shulchan Arukh, Orach Chaim 299:10) suggests that since the Gemara (Pesachim 104a) says that the first water refers to the lower waters, which are less holy than the higher waters. Once you start with the less holy, you do not need to use a double "bein."

R. Yitzchak Meir Goodman (Great Torah Lights from Great Torah Minds on Lev. 11:47) points to four other places in the Torah where a distinction is made with only one "bein" -- Lev. 20:25, 27:33, Num. 26:56, Deut. 17:8. However, those cases do not use the verb "le-havdil" to make a distinction. It could be that only "le-havdil" functions in that fashion.