Our gemara explains why, it doesn't have a different Korban Musaf. However, the Medrash (quoted l'halacha) has a different reason, that since the מצרים drowned we don't say Shira. In fact, the Maharsha in Sanhedrin (39b) asks why do we need the reason of our Gemara, the reason of the medrash is better (עיי"ש).
We can certainly ask why did the medrash give a different reason then the Gemara and why do the Poksim quote the medrash?
The answer could be based on the end of the sugya. The Gemara asks why do we say Hallel on Chanukka (it doesn't fit the criteria outlined above) and the gemara answers because a נס happened. Based on this we could say as follows. You don't say Hallel on the seventh day of Pesach מטעם יו"ט because of the Gemara's reason that it is not חלוק בקרבנות, however, we should say hallel because of the נס of קריעת ים סוף. By the hallel of a נס the seventh day of Pesach should seemingly qualify (it is no worse then Chanuka). Therefore we need the reason of the medrash which is a reason why we don't say hallel on the נס on the seventh day of Pesach.