Rabbi, as in my Rav, is referring to ones teacher, their 'Moreh Hora'ah'. The one who instructs them on what to do in practice.
Rebbe is a Roshei Teivot for 'Rosh B'nai Yisroel', 'Head of the Jewish People' as in parshat Ki Tisa (Shemot 30:12). This is also brought by Rabbi Nachman Goldschmidt of Techerin in Parparot l'Chochmah, section 111.
In midrash and other places, the Jewish people are compared to a single individual, like in the concept of כל ישראל ערבים זה לזה. This same concept is why for example we say Viduy in the plural instead of the singular and confess to things that we may not be individually guilty of committing. Rabbi Moshe Cordevero explains that confessing to something you have no connection to is a false confession and is prohibited. But because the Jewish people are all one, spiritually and physically, we can make the entire Viduy truthfully.
Each individual corresponds to a particular limb or part of a limb. This idea can also be seen in parshat Shoftim (Shemot 18:21,25) where the judges are also referred to as 'head' and are compared to the eyes of the community (עיני העדה). In the commentaries to this citation, it explains that these "judges" are referring to the establishment of the Sanhedrin which is headed by the Nasi.
In the specific case of the individual called Rebbe, it is the idea of the complete head, from which life flows out to all the other limbs. This follows the explanation of Rabbi Yosef Caro in Maggid Mesharim on parshat Ki Tisa and also Yalkut Shimoni on Tetzaveh 376.
And this concept is also associated with the idea of the term 'Nasi', which is referred to in Aramaic in the case of the tribes as 'אמרכל' which translates as 'speaking for the whole'.
Nasi, like Rebbe, is another Roshei Teivot meaning 'Nitzutz shel Yaacov Avinu', a spark of Yaacov avinu. The Lubavitcher Rebbe brings this from Sefer Kehilat Yaakov in his talk from the 10th of Shevat, 5722, which can also be found in Likkutei Sichot, volume 4, page 1051, note 18.
Sefer Kehilat Yaakov by Rabbi Yaakov Tzvi Yolles of Dinov explains the origin of the word "Rebbe" and clarifies why they referred to Rebbe, meaning Rabbi Yehuda HaNasi as Nasi. He associates the title of Rebbe to that of Nasi and explains that the source of this title (Nasi) is because he is a spark of Yaakov Avinu (נצוץ של יעקב אבינו).
The Lubavitcher Rebbe goes on to explain that the Kehilat Yaakov is basing this upon what is found in Bava Batra 58a which compares the beauty of Yaakov Avinu to that of Adam HaRishon. Just as all mankind came from Adam HaRishon, so too all the tribes of Israel came from and were connected to Yaacov, and so too all the Jewish people are connected to the Nasi of their generation.