The source for counting Sefirat HaOmer is found in Sefer VaYikra (23:15): “ּU’Sefartem Lachem MiMachorat HaShabbat MiYom Haviachem Et Omer HaTenufa, Sheva Shabetot Temimot Tihiyena,” “And you shall count for yourselves, from the day after the rest day, which is the day you bring the Omer as a wave offering, seven weeks; they shall be complete.” The words “MiMachorat HaShabbat” are the source for the timing of the start of the Omer, and since the phrase literally means “the day after the Shabbat”, the interpretation of the pasuk was subject to a debate between the Perushim (i.e. the Chachamim) and the Tzedukim. The Gemara in Menachot (65a- 66a) relates that the Tzedukim took the phrase at face value and ruled that the counting of the Omer begins on Saturday night, and hence Shavuot would always fall out on a Sunday. The Perushim, however, interpreted that the phrase meant the second day of Pesach, and so the count begins on the night of the second day, because “Shabbat” can simply mean “holiday” in some contexts. Hence, Shavuot would always fall out on the same day in Sivan.
Why was the Torah not more clear in mandating which day the Omer begins from? Rav Yosef Dov Soloveitchik answered this question by dealing with another question. Before introducing the Omer, the Torah discusses all of the Inyanei Pesach and only then discusses the concept of Sefira. The Torah should have mentioned the Omer when it discussed the second day of Pesach; why did the Torah push off the discussion of the Omer until after the laws of Pesach? The Rav proposes that this separation of these two laws in the Torah may indicate there is no true connection between Pesach and Omer besides for the date. That is why the Torah referred obliquely to the second day of Pesach and instead called it “Machorat HaShabbat.”
We can understand this apparent distinction between Pesach and Omer by turning to the inner meaning of both. The original Pesach was a time when not only every Jew, but also every non-Jew, believed in Hashem. Open miracles were happening all the time, such as the ten plagues and the splitting of the sea. Even when the Jews reached the barren desert, there were still open miracles occurring, such as the Ananei HaKavod and the Man, which fell daily from the sky. The question was, what would happen when Hashem would stop doing open miracles? The Jews would go into Israel and not be able to rel ompletely on Hashem for sustenance and protection -- would they still believe in Hashem? That is the point of the Korban HaOmer. The Korban HaOmer consisted of the first barley harvested and was given to the Kohen. It ensured that each farmer would remember that it was Hashem who made the crops grow and gave him his bounty, and not “Kochi Ve’Otzem Yadi,” “the strength of my hands.”
Therefore, it becomes quite evident why the Torah deliberately completely cut off all connection to Pesach when discussing the Omer. Whereas Pesach embodies a total reliance on Hashem that occurred immediately after Yetziat Mitzrayim, the Omer signifies the situation in which we find ourselves today-- working for our subsistence but still needing to recognize that everything comes from Hashem.