There is a very interesting historical connection between these two dates, which very few people notice.
The second of Adar (598 BCE) was the date that the First Temple was conquered by Nebuchnazzaer, and King Jeconiah taken captive.
This comes from Chronicles of Chaldaean Kings (626-556 B.C.) in the British Museum.
“In the seventh year, the month of Kislev, the king of Akkad mustered his troops, marched to the Hiatti-land, and encamped against (i.e. besieged) the city of Judah and on the second day of the month of Adar he seized the city and captured the king.”
And then, according to tradition exactly 70 years and one day later, on the third of Adar (515 BCE), the Second Temple completed
Ezra 6:15
וְשֵׁיצִיא בַּיְתָה דְנָה, עַד יוֹם תְּלָתָה לִירַח אֲדָר--דִּי-הִיא שְׁנַת-שֵׁת, לְמַלְכוּת דָּרְיָוֶשׁ מַלְכָּא.
And this house was finished on the third day of the month Adar, which was in the sixth year of the reign of Darius the king.
So the Jewish people slept for 70 years (as the story of Choni the circle-drawer sleeping for 70 years highlights), then woke up the next day and rebuilt the Temple.
The mishna in Ta'anit says that when Adar begins we should increase our happiness. It is hard to reconcile the destruction of the Temple with the happiness of Adar. But if the destruction only lasted for a day, perhaps that is reason to celebrate.
