The recent Tu B'Av mini-holiday led me to thinking: the Gemara in Taanis says that the two happiest days of the year for the Jewish people in Temple times were Yom Kippur and Tu B'Av. For the former, it's obvious that a day that culminated in national cleaening of sins committed would be an occasion of joy. But Tu B'Av? Yes, the gemara lists a bunch of good things that happened on that day but it has never really achieved the prominence that Yom Kippur has.
But if the gemara describes the two in similar fashions, there must be an underlying connection. Here is my suggestion.
First, one must remember the primary reasons for a day being considered special in Jewish tradition. For example, Tisha B'Av has become a national day of mourning for pretty much every tragedy that has befallen us since the destruction of the First Temple (may it speedily be rebuilt). But according to rabbinic tradition, the original appearance of Tisha B'Av is in the Torah. In a famous aggadah, Chazal notes that the day that the meraglim reported their slander about Eretz Yisrael and caused the nation to cry in despair was Av 9. As is famously known, they reported that God looked down and say "Tonight you are crying for no reason. I'll give you a reason to cry on this night for generations."
In this vein, let's look at the first appearance of Yom Kippur in the Torah. Contrary to popular belief, it isn't in the various lists of holidays in the latter part of Shmos or in the description of the Kohen Gadol's ritual in Acharei Mos. Rather, it's actually the real day of matan Torah.
Remember that according to tradition, Moshe Rabeinu, a"h, received the first set of luchos from God on Sivan 6 or 7 (depending on who you hold by). However, these tablets didn't last long. They didn't even make it to the camp of our ancestors before being smashed because of the sin of the golden calf. This is perhaps one reason we call Shavous z'man matan Toraseinu and not z'man kabalas Toraseinu. God gave over the Torah on that day but we didn't actually receive it.
When did we actually receive it? Well following the chronology, it was when Moshe Rabeinu returned to the camp after his third 40 day stay in Heaven. Given that his descent the first time was on Tammuz 17, Chazal calculate that this final descent when the Torah made it to our ancestors was on Tishrei 10, Yom Kippur. This Yom Kippur is, in a quiet way, z'man kabalas Toraseinu.
Tu B'Av, on the other hand is noted by Chazal to be the day the dor hamidbar who were condemned to wander for 40 years finished dying off. Recall that after the meraglim were punished, God announced that anybody over 20 years old would finish their lives in the midbar and only when they were all gone would our ancestors be allowed to enter Eretz Yisrael. Thus after Tu B'Av the final march towards our Land could begin.
What's the connection then between Yom Kippur and Tu B'Av? When God first reveals Himself to Moshe Rabeinu, He tells him that the upcoming Yetzias Mitzrayim will have two objectives. The first is to go to Har Sinai and receive the Torah. The second is to take our ancestors to the Land promised to our Forefathers, the land of Israel.
In both cases, there were complications. The receiving of the Torah was delayed by over 80 days due to the sin of the Golden Calf. The final march to Eretz Yisrael was held up by 38 years.
Thus both holidays share an important feature in common - the achieving of one of the goals of Yetzias Mitzrayim. Together then they testify to God's redeeming us from slavery and bringing us as the people of Torah to the land of Torah.