Rabbi Yehudah the son of Rabbi Shimon ben Pazi said: Dovid said one hundred and three chapters of Tehillim and he did not say Hallelu'kah (Praise God) until here. This is because of the verse preceding this one: The sinners will cease from the earth and the wicked will be no more. It is only then that he could proclaim, Hallelu'kah, because then he saw the downfall of the wicked.
The Maharsha asks from that which the Beis Yosef (O"C 490) brings in the name of the Medrash that the reason why we do not complete Hallel on all the days of Pesach is because the Egyptians were drowning in the sea and it is written: When your enemies fall, do not rejoice. The Gemora in Megillah (10b) states: Rabbi Yehoshua ben Levi introduced his lecture on Megillas Esther with the following interpretation: It is written [Devarim 28:63]: And it shall come to pass that as Hashem rejoiced over you to do you good, and to multiply you, so will Hashem rejoice to destroy you. Does Hashem rejoice when the wicked are in misfortune? Rabbi Yochanan states that Hashem does not rejoice at the downfall of the wicked. How could Dovid sing a song when he witnessed the downfall of the wicked?
The Maharsha asks furthermore: The Gemora in Eruchin (10b) offers a different reason as to why we do not complete Hallel on all the days of Pesach. It is because the korbanos offered are identical each day (unlike Sukkos, where there are different korbanos brought every day). What is the necessity of the other reason?
The Meon Habracha suggests the following answer: There are two reasons for the obligation to recite Hallel; one is because the day is a Yom Tov, such as Sukkos and Shavuos. There is also an obligation to recite Hallel on account of a miracle that happened to us. Both reasons should apply on Pesach. It is a Yom Tov, and a miracle occurred. The Gemora in Eruchin is discussing why the complete Hallel is not recited every day of Pesach because it is a Yom Tov. On that, the answer is because the korbanos offered each day are identical. The Medrash is asking that a complete Hallel should be recited on the last day of Pesach because of the miracle that happened to us by the splitting of the sea. On that the answer is that it is not proper to sing when your enemies have fallen.