Why do some of the Israeli Left (politically) argue that they believe in the teachings of Rabbi Yochanan ben Zakkai over the teachings of Rabbi Akiva?
Shortly before the year 70, Jerusalem was surrounded by Roman troops. Raban Yochanan ben Zakai attempted to negotiate a surrender with the Romans. He recognized there was no realistic outcome in which Jewish self-rule remained over Jerusalem. He was opposed by Jewish religious terrorists known as the Sicarii (Latin for "dagger people") who wanted to force what they believed would be an apocalyptic war against the Romans.
The fall of Jerusalem is described in the Talmud in Gittin, around page 56a. Here's a key passage:
הוה להו למיזן עשרים וחד שתא הוו בהו הנהו בריוני אמרו להו רבנן ניפוק ונעביד שלמא בהדייהו לא שבקינהו אמרו להו ניפוק ונעביד קרבא בהדייהו אמרו להו רבנן לא מסתייעא מילתא קמו קלנהו להנהו אמברי דחיטי ושערי והוה כפנא
Jerusalem had enough stores of food to withstand a siege for years. But there were Biryonei in the city. [Rashi: Biroynei are "empty people who rush to war." Let's translate "militants."] The rabbis wanted to go out and make peace with the Romans, but the militants wouldn't let them. The militants wanted to go wage war against the Romans, but the rabbis told them it wouldn't succeed. So the militants got up and burned Jerusalem's food stores, and then there was famine.
A few lines further down:
אבא סקרא ריש בריוני דירושלים בר אחתיה דרבן יוחנן בן זכאי הוה שלח ליה תא בצינעא לגבאי אתא א"ל עד אימת עבדיתו הכי וקטליתו ליה לעלמא בכפנא א"ל מאי איעביד דאי אמינא להו מידי קטלו לי א"ל חזי לי תקנתא לדידי דאיפוק אפשר דהוי הצלה פורתא
"Daddy Dagger", the head of the Jerusalem militants, was the nephew of Chief Rabbi Yochanan ben Zakai, and secretly met with the rabbi. "How much longer can you keep doing this and kill everyone by famine?" -"What can I do, Uncle Yochanan? [The militants call me the leader but] if I say anything, they'll kill me too!" "See if you can arrange for me to get out of Jerusalem [and negotiate], maybe I can still save something."
Around the year 130, Rabbi Akiva supported Bar Kochba's revolt against the Romans in Beitar. The revolt stood a good chance of succeeding strategically, and thus Rabbi Akiva felt that this could in fact be a final war in which Bar Kochba could be the Messiah.
In broad strokes, those who believe in negotiated surrender will point to Raban Yochanan ben Zakai's actions, and those who believe in conflict will point to Rabbi Akiva's words.
The truth is, lo and behold, both could be right: you fight when you have a fighting chance, and don't when you don't. Rabbi Hershel Schachter shlit'a, for instance, has remarked that the Talmud intentionally omits the Masada conflict because they fought when there was no rational chance of winning. His approach to the question of land-for-peace, as appeared in the Journal of Halacha & Contemporary Society in the 1980s, is that two questions must be asked: what are the rational chances that armed conflict will succeed in a non-miraculous way, and what's the definition of "success"?
The question then becomes how to assess a given situation ... and that's beyond the scope of this site.
The other question becomes what methods are used; the Talmud is making it quite clear that by forcing a famine and killing moderate voices, the Sicarii were automatically in the wrong. (Bar Kochba himself goes south when he gets paranoid and kills a rabbi based on a flimsy rumor that he's a Roman sympathizer.) As Elie Wiesel has put it: "irrespective of whether theoretically the ends justify the means -- if you employ certain means, that fundamentally changes the ends."