What is the difference between "ani" and "anachi" in biblical Hebrew? They both mean "I", so why are there two different words that mean the same thing?

Shalom, Thank you for your interesting question. We see in many places that the Torah uses the words Ani and Anochi (both of which mean "I") with an apparent subtle difference. There must be a difference between Jacob saying "Anochi - I am Esau your first born son", as opposed to Esau's "Ani - I am Esau" (Bereshit 27,19 and 32). Also, in the Ten Commandments, G-d uses the word "Anochi - I am the Lord your G-d", but elsewhere the term "Ani - I am the Lord your G-d" is used, as in the end of Shema. So your question is a good one. Many commentators have noted the difference and I will quote just two (in the hope that they will encourage you to continue your research into this question with further Torah study). The Malbim writes (Bereshit 27,18) that "Anochi" is used when referring to oneself as opposed to anyone else - I and no other. But "Ani" is used whenever one talks about themselves, even when not coming to exclude someone else specifically - I ate an apple. Rabbi S. R. Hirsch (Shemot 20,2) writes that "Ani" is used to express the person speaking and not someone else, but "Anochi" is used when the speaker wants to express a desire to get close to the person they are addressing. I hope this will be a good start for you to continue to find other explanations that will open up the depth and beauty of the Torah for you. Blessings.