The Talmud in Yevamot 62B:
ר"ע אומר: למד תורה בילדותו ילמוד תורה בזקנותו; היו לו תלמידים בילדותו יהיו לו תלמידים בזקנותו. שנא' " בַּבֹּ֨קֶר֙ זְרַ֣ע אֶת־זַרְעֶ֔ךָ [ וְלָעֶ֖רֶב אַל־תַּנַּ֣ח יָדֶ֑ךָ כִּי֩ אֵֽינְךָ֨ יוֹדֵ֜עַ אֵ֣י זֶ֤ה יִכְשָׁר֙ הֲזֶ֣ה אוֹ־זֶ֔ה וְאִם־שְׁנֵיהֶ֥ם כְּאֶחָ֖ד טוֹבִֽים׃ ] "
אמרו: שנים עשר אלף זוגים תלמידים היו לו לרבי עקיבא מגבת עד אנטיפרס וכולן מתו בפרק אחד מפני שלא נהגו כבוד זה לזה. והיה העולם שמם עד שבא ר"ע אצל רבותינו שבדרום ושנאה להם: ר"מ ור' יהודה ור' יוסי ורבי שמעון ורבי אלעזר בן שמוע - והם הם העמידו תורה אותה שעה.
תנא: כולם מתו מפסח ועד עצרת.
אמר רב חמא בר אבא, ואיתימא ר' חייא בר אבין: כולם מתו מיתה רעה.
מאי היא? א"ר נחמן: אסכרה.
I have two questions on this piece. The Talmud's reason for the death of R' Akiva's 12,000 pairs of students is very difficult to understand: why the extreme punishment for not giving each other כבוד, respect/honor? Additionally, what is the significance of the fact that they all died in one period of time, between Pesach and Shavuos?
The first thing that must be understood is why people honor (or do not honor) others? Giving one's fellow honor is a way of recognizing his authority, power or mastery - if not over you then at least over some one or thing. By not giving someone honor you are implying the opposite - i.e., that person has no authority or power. Certainly, all of R' Akiva's students were worthy of honor - they had the authority that comes along with knowledge. So why would his students not be respectful to each other? Possibly, they viewed their Torah knowledge solely as a means to power and prestige - to give honor to each other would have detracted from each individual's pursuit of power.
One of the ultimate lessons of Yetziat Mitzrayim was that man is utterly lacking in power before G' and completely dependent on Him. G' revealed Himself to us in Mitzrayim as the sole possessor of power, only He could take us out. This is what G' told Moshe to tell the people (Sh'mot 6):
ו לָכֵ֞ן אֱמֹ֥ר לִבְנֵֽי־יִשְׂרָאֵל֮ אֲנִ֣י יְהוָה֒ וְהֽוֹצֵאתִ֣י אֶתְכֶ֗ם מִתַּ֨חַת֙ סִבְלֹ֣ת מִצְרַ֔יִם וְהִצַּלְתִּ֥י אֶתְכֶ֖ם מֵעֲבֹֽדָתָ֑ם וְגָֽאַלְתִּ֤י אֶתְכֶם֙ בִּזְר֣וֹעַ נְטוּיָ֔ה וּבִשְׁפָטִ֖ים גְּדֹלִֽים׃ ז וְלָֽקַחְתִּ֨י אֶתְכֶ֥ם לִי֙ לְעָ֔ם וְהָיִ֥יתִי לָכֶ֖ם לֵֽאלֹהִ֑ים וִֽידַעְתֶּ֗ם כִּ֣י אֲנִ֤י יְהוָה֙ אֱלֹ֣הֵיכֶ֔ם הַמּוֹצִ֣יא אֶתְכֶ֔ם מִתַּ֖חַת סִבְל֥וֹת מִצְרָֽיִם׃ ח וְהֵֽבֵאתִ֤י אֶתְכֶם֙ אֶל־הָאָ֔רֶץ אֲשֶׁ֤ר נָשָׂ֨אתִי֙ אֶת־יָדִ֔י לָתֵ֣ת אֹתָ֔הּ לְאַבְרָהָ֥ם לְיִצְחָ֖ק וּֽלְיַעֲקֹ֑ב וְנָֽתַתִּ֨י אֹתָ֥הּ לָכֶ֛ם מֽוֹרָשָׁ֖ה אֲנִ֥י יְהוָֽה׃
(In fact, B'nei Yisrael played no active role in their redemption - they were redeemed on the merit of demonstrating their subservience to G' and their rejection of the idols of Mitzrayim through the korban Pesach.) This breakthrough, in B'nei Yisrael's relationship to G', was developed and expanded over the next seven weeks (through their experience at the yam suf, at mara, the mahn, the battle with Amalek, Shloshet y'mei hagbala etc., ואכמ"ל). On the fiftieth day, they were ready to receive G's Torah, His Guidance, at Har Sinai. At Sinai, B'nei Yisrael expressed their absolute subservience to G' through accepting and following His Laws. By that point they had no illusions about who possessed real power. For them the Torah was a pure expression of Malchut Shamayim - they could not, at that point, distort the Torah and turn it into a vehicle for human glorification.
Every link in the chain of the Mesora must make the Pesach breakthrough or risk distorting the Torah. The Torah can not be received - and certainly not transmitted - by one who views it as a tool for personal glorification, as did R' Akiva's students. Torah must be a pure expression of G's Malchut - not malchut Adam. This idea is most clearly expressed in Pirkei Avot (4:7):
רבי צדוק אומר, לא תעשם עטרה להתגדל בהם, ולא קורדום לחפור בהם: כך היה הלל אומר, ודישתמש בתגא חלף; הא, כל הנהנה מדברי תורה, נטל חייו מן העולם.
Rabbi Tzadok said: Do not make the Torah a crown with which to make yourself great, nor use it as a shovel with which to dig. As Hillel used to say: He who makes worldly use of the crown of the Torah shall perish. Therefore, any one who benefits from the words of the Torah takes his life from the world.
This lesson is of the utmost importance to reflect on between Pesach and Shavuos - the time when our ancestors prepared themselves for kabbalat haTorah.. This is the significance of R' Akiva's students dying during this time period - as if to say that on account of their failure to incorporate the lessons of Pesach they died. They did not respect each other because their Torah was self serving not a means to serve G'. It is hoped that by mourning them we inspire ourselves to not make their mistake and instead learn Torah as an expression of our acceptance of G's malchut.