Notice that the 10 Commandments begin "Anokhi Hashem Elokekha", while Shema ends "Ani, Hashem Elokeikhem." And how Yaakov says "וַיֹּאמֶר יַעֲקֹב אֶל אָבִיו, "אָנֹכִי עֵשָׂו בְּכֹרֶךָ עָשִׂיתִי כַּאֲשֶׁר דִּבַּרְתָּ אֵלָי (Bereishis 27:19) While the real Esav says וַיֹּאמֶר לוֹ יִצְחָק אָבִיו "מִי אָתָּה?" וַיֹּאמֶר "אֲנִי בִּנְךָ בְכֹרְךָ עֵשָׂו." (Ibid. v. 32)
Therefore commentators were motivated to look at the difference. But this is in terms of connotation and derashah, not translation.
Rav SR Hirsch (Shemos 2:20, on the first commandment) says that "ani" refers to the person in contrast to other people, whereas "anokhi" refers to the person in relation to the listener. The Malbim (on Bereishis 27) and similarly Shadal write that "ani" is the normal word for "I", whereas "anokhi" is used to emphasize "and not others".
So in general, "ani" would mean that the emphasis of the phrase in on the who, whereas "anokhi" would emphasize what it was that I did or that happened to me.
I think all of these are consistent with a derashah R' Yochanan makes on Shabbos 105a on the "Anokhi" at the start of the Commandments. He says Hashem used "Anokhi" as an acronym for "אנא נפשי כתיבת יהבית -- I Myself [lit: My Soul] made this writing." Generalize that to what you think "Anokhi" connotes across the board, and you have the kinds of ideas we see above.
(In the Zohar, it says that Hashem's "Ani" refers to the Lower Throne [Malkhus as it is manifest within creation], and His use of "Anokhi" refers to the Upper Throne [comprised of Binah as connected to Malkhus]. I have no idea what that means.)