The 20th of Sivan is a day that was twice designated as a fast day for massacres against European Jewry; once by Rabbeinu Tam and once by the Shach. Special Selichos were said on this day. Below is a recent question and answer from the Ask Revach page that adds some light on the subject and the minhag today.
Question submitted by Chaim: Why were chazal mesaken [decree] chof [20th] Sivan as a day of mourning for ta'ch v'ta't [massacres during Bogdan Chmielnicki’s Cossak uprising] and yet there is so much controversy over Yom Hashoah? Additionally, why has chof sivan, the day of morning for ta'ch v'ta't, fallen out of favor?
Answer by Rav Peretz Moncharsh: I think that a little background information will put things in their proper perspective.
The 20th of Sivan was first designated as a fast day by Rabbeinu Tam after the first Jews were executed because of the accusations of a blood libel. Observation of the fast gradually faded over the ensuing years as that tragedy was overshadowed by the 150 years of the Crusades.
After the [Chmielnicki] massacres of Tach v'Tat when approximately a third of European Jewry was killed, the fast of the 20th of Sivan was reinstituted, as on that date the glorious Jewish community of Nemirov was destroyed by the Cossacks. This decision was made by the Shach and the Vaad Arba Aratzos, and was confirmed by the Shela HaKadosh, Tosafos Yom Tov, Magen Avraham and many other Gedolim.
I'm not sure why it has faded, but probably the following 300 years provided more than their fair share of tragedies to eclipse Tach v'Tat. Also, the enactment seems to have been made specifically for the Jews living in the Polish kingdom.
Yom HaShoah never had the support of a wide array of Gedolim, and in fact the Rabbanut even designated the 10th of Teves as the most appropriate day to commemorate the Holocaust. Furthermore, the date of Yom HaShoah was chosen to commemorate the valor of those who participated in the Warsaw ghetto uprising, and not the tragedy of the 6 million who were murdered. Additionally, Yom HaShoah is in Nissan, a month during which according to halacha we do not engage in public mourning.