The Talmud relates (BT Shabbat 31):
“A gentile once came to convert to Judaism, on the condition that he could learn the whole Torah while standing on one foot. He approached Shammai, who rejected him. So he went to Hillel and asked him that he wanted to be taught the Torah while standing on one foot. Hillel told him: “That which you hate, do not do to your neighbor. That is the whole Torah, and everything else is commentary–go and learn!”
Another gentile who wanted only the written Torah came to convert. Shammai refused him, so he went to Hillel. The first day, Hillel taught him the Hebrew alphabet. However, the next day Hillel reversed the order of the letters. Confused, the convert asks, “But yesterday you said the opposite!?” Said Hillel: “Now you see that the written word alone is insufficient. We need the Oral Tradition in order to understand the Written.”
A third gentile was very impressed by the Priestly garments and came to convert. Again, Shammai dismissed him, but Hillel encouraged him to study more. After learning, he came to realize that even David, King of Israel didn’t qualify to serve as a priest in the Temple, because he wasn’t born a cohen.” The latter came back to Hillel and said, ‘O gentle Hillel; blessings rest on thy head for bringing me under the wings of the Shekhinah!’ Some time later the three met in one place; said they, Shammai’s impatience sought to drive us from the world, but Hillel’s gentleness brought us under the wings of the Shekhinah (BT Shabbat 31a).
Hillel’s liberal philosophy endeared him to his entire generation because he personified kindness, patience, and a love for all people. Shammai’s machmir (stringent) philosophy, you might say, had a repelling effect. He probably could have benefited from Dale Carnegie’s “How to Win Friends and Influence People,” but self-help books did not exist in Shammai’s time.
Rabbinic tradition acknowledges the Halacha follows Hillel’s liberal approach to the conservatism of Shamai (Soferim xvi. 9). It is fair to ask, why would the Halacha follow Hillel–the liberal–over Shammai, who was notoriously known for his ultra-conservative positions? The answer is simple; Hillel behaved like a mentsch at all times.
The Jerusalem Talmud records a story where Hillel and Shammai once differed regarding the ritual status of grapes and olives. As with most arguments, the issue is not what either party is arguing about, it is rather, how and why they are arguing. As the controversy grew increasingly hostile, it appears that the Shammites turned violent against the Hillelites. To calm the crowd, Hillel showed his greatness by de-hostilizing the mob when he assumed a subservient position. Without fanfare, Hillel proved why he was so morally superior to his adversary, who probably viewed Hillel’s subservience as a “moral victory. ” Shammai’s abuse of his position was nothing less than idolatrous.
Jealousy isn’t always bad. Jealousy can often serve a positive function in motivating students to improve. Aristotle once wrote in his “Art of Rhetoric,” where he explains, “Jealousy is both reasonable and belongs to reasonable men, while envy is base and belongs to the base, for the one makes himself get good things by jealousy, while the other does not allow his neighbor to have them through envy.” 
Shammai’s personal disdain (or might it have been envy?) for Hillel grew so intense that “A sword was planted in the House of Study and it was proclaimed, ‘He who would enter, let him enter, but he who would depart, let him not depart!’ And on that day Hillel sat submissive before Shammai’s rulings, like one of the disciples, and it was as grievous to Israel as the day when the golden calf was made ….”  Imagine how one Jew turned against the other, as the Shammaites attempted to murder the Hillelites for disagreeing with their master. This was the day when one brother killed another brother–all for the sake of Halacha?!
Eventually, a Heavenly Voice made Her Presence known at Yavneh, decades after the destruction of Judea and Jerusalem, which ruled that the Halacha follows the Academy of Hillel in every instance.
And now you know–the rest of the story!
 Aristotle, “The Art of Rhetoric,” sect. 6, ch. 2.11.
 TB Shabbat 17a; TJ Shabbat 1:4, 9a (1:8, 3c); Tosefta Shabbat 1:16. See also Lieberman, Tosefta ki-Feshutah, Shabbat, p. 15, for clarification of the statement in TB Shabbat, “On that day Hillel sat bent over [i.e., subservient] before Shammai,” which implies that even in the days of Shammai and Hillel themselves, relations between them were difficult. This statement does not appear in the other sources. See Tosafot, Shabbat 14b, s.v. Veillu Shammai ve-Hillel; S. Lieberman, Yerushalmi ki-Feshuto, pp. 38, 52. This is no great surprise for the reasons mentioned above.
 Tosafot, Bava Meẓiʾa 59b, s.v. Lo va-shamayim hi.