There are two interesting questions about the Yom Tov of Shavuos that set it apart from the other Yomim Tovim. First, there is no particular mitzvah or commandment to be done on this Yom Tov (unlike Rosh Hashanah which has Shofar, Pesach which has Matzah, or Sukkos which has Sukkah and arba minim). Second, the Torah does not give a specific date for the celebration – rather it is counted from the second day of Pesach.
The Bnei Yissaschar addresses the question of the date of the holiday with the following idea. Positive mitzvos in the Torah usually have a specific time when they are supposed to be performed. In regards to the mitzvah of learning Torah, the Torah makes special mention that one must toil in the Torah both day and night, as if to say that one is obligated to learn all the time. He explains this that the Torah is teaching us that the realm of learning Torah is on a plane where time does not exist. Thus, the mitzvah of learning Torah does not have a time.
I found the same idea in other seforim in regards to why there are no specific mitzvahs for the holiday of Shavuos. Every mitzvah relates to a certain connection that we have to Hashem. The holiday of Shavuos is about our basic connection to Hashem, achieved through learning. Therefore, it would be inappropriate to accentuate a certain aspect being that the holiday relates to all aspects of our relationship. This was the outcome of the idea “and they all came with one covenant, together.”
I would like to add another thought on this subject. We find in the mitzvah of sefiras ha’omer the Torah relates to the counting as a counting of fifty days. Yet in reality we know that we only count 49 days! There are various answers given to this question, but the bottom line is that Shavuos defies all logic. We cannot pin it down with a mitzvah or a time. This is even more apparent as the date of receiving the Torah was on the seventh of Sivan, and yet we celebrate Shavuos on the sixth of Sivan! All of this comes together to bring out the idea that all aspects of the Torah cannot be quantified or qualified through the human mindset.
There is a very popular custom to stay up on Shavuos night, which seems quite counter-productive. One would expect to get sleep so that one can use the day properly. Likewise, one would expect to have meals with rich foods of beef and chicken, as we do on other yomim tovim. Yet many people have the custom of eating dairy foods.
This all reminds us that we said “na’ase v’nishma” and accepted upon ourselves to live on a plane that we cannot humanly understand. We therefore celebrate Shavuos on a totally different plane.