There are three reasons why we fast on Tannis Esther, and there is a practical difference between this fast and all the other fasts.
The Tur (686:2) says the custom is to fast on Tannis Esther, the 13th of Adar. This custom is mentioned in Tosafos (Tannis 18a) and Rosh (Megillah 1:8).
Although there is no direct source in Tanach or Talmud to it, the Rosh (Megillah 1:1) and the Ran (IBID 2b) both quote Rabbenu Tam that suggests when the Gemarah (2a) says "the thirteenth of Adar is a day of assembly for all," it refers to assembling to recite selichos because it's a fast day. They fasted on the 13th of Adar for success during the war being fought on that day.
According to Rabbenu Tam, Tannis Esther is commemorating the fast they observed on the 13th of Adar. The purpose of the fast is to daven, and remember that Hashem listens to all our prayers. This is the classic approach, as mentioned in the Mishnah Berura (686:2).
However, Kol Bo (end siman 62) says Tannis Esther is commemorating a different event: Esther instructed Mordechai (4:16) to gather all the Jews, and they should fast for three days so that she should be successful when going to the king.
The Yesod VeShoresh HaAvodah (12:3) gives a third reason why we fast: Anytime a miracle happens to us, we are obligated to fast and thank Hashem for the miracle each year on that day. Since we were saved from our enemies during the Purim story, we fast to thank Hashem.
Based on the above three reasons (fast of the 13th, three-day fast, fast to thank Hashem), this fast differs from all the others. The other fasts commemorate a sad event connected to the destruction of the Bais Hamikdosh. Tannis Esther, however, is not linked to anything sad, the Kol Bo (IBID) even suggests it's a day of simcha.
R' Shmuel Kamenetsky (Kovetz Halachos, Purim 2:11) states because Tannis Esther is not commemorating anything sad, there is a practical difference between this fast and all others. The Mishnah Berura (550:6) says that a "ba'al nefesh" (someone who cares about his soul) keeps the restrictions of Tisha Bav- not showering, etc., on all the fasts. However, even those who refrain from showering during the other fasts still could shower on Tannis Esther because it's not a sad day.