Yetzias Mitzrayim, which we celebrate on Pesach, is significant, among other reasons, because it became the prototype for all future redemptions. Hence, in many of the later miraculous victories of the Jewish people, comparisons are made with the events of Yetzias Mitzrayim. For example, when the Assyrians put a siege around Yerushalayim, the Novi stressed that an angel killed the Assyrian army in the night, comparing the event to Makkas Bechoros, the tenth plague in Egypt (מלכים ב' י"ט:ל"ה). This comparison actually prompted Chazal to claim that that redemption occurred as well on Pesach night. In cases such as this one, the comparison is positive, because the redemption that occurred really was reminiscent of what happened in Egypt. The Assyrian minister Ravshakeih claimed that Hashem was worthless, much like Paroh had done before him. Chizkiyahu told the people that Hashem would save them, but as the Assyrian threat got stronger, Chizkiyahu, like Moshe before him עיין שמות( ה':כ"ב-כ"ג), started getting nervous about when Hashem was going to rescue the Jews. Yeshayahu reassured him not to worry, and eventually Hashem brought a plague reminiscent of the Makkas Bechoros which punished the Egyptians. While one could argue that the redemption from Assyria wasn't as great, because it was done by a angel and not by Hashem Himself (as the Mechilta that is cited in the Haggadah stresses was the case regarding Makkas Bechoros), the people in the generation of Chizkiyahu most likely felt that they had experienced a new Yetzias Mitzrayim in their own time.
The Assyrian defeat was compared to Yetzias Mitzrayim because it happened in the middle of the night. Other Jewish victories were compared to Yetzias Mitzrayim in other ways. One such comparison relates to the response of Shirah, a song of celebration following triumph. There are two songs that are clearly compared to Shiras HaYam, sung at the Red Sea following the culmination of Yetzias Mitzrayim, by means of the format in which they are written, specifically, the blank spaces in between each word, namely, Shiras Devorah and Shiras Dovid. Interestingly, Shiras Devorah is the Haftorah read when we read Shiras HaYam on the Shabbos of Parshas Beshalach during the year, while Shiras Dovid is the Haftorah read when we read Shiras HaYam on the seventh day of Pesach. Why the difference? We must take a look at the events surrounding each song.
Devorah and Barak's victory )שופטים פרק ד'( came about in the war against the Canaanim from Chatzor. It was a purely natural miracle, in which every soldier died in the battle except for Sisra, the general (which reminds us of Chazal's comment about how Paroh was the only surviving Bechor in Egypt and thus later saw his entire army drown in the Yam Suf). The battle had no fireworks; no angels were flying around killing enemy troops. The battle, albeit a great victory, was no Yetzias Mitzrayim; it had no ten plagues or miracles like Kerias Yam Suf. Although the people were undoubtedly ecstatic about the victory over Chatzor, Devorah, in her prophetic wisdom, alluded to this fact that the victory was less than equivalent to Yetzias Mitzrayim at the end of her song. In contrast to Shiras HaYam, which ends with the Posuk saying "ה' ימלך לעולם ועד," emphasizing Hashem's eternal reign (שמות ט"ו:י"ח), Shiras Devorah ends with the Posuk saying "ותשקט הארץ ארבעים שנה," indicating that after the war there was peace for forty years (שופטים ה':ל"א), but then, Eretz Yisrael was reoccupied, this time by Midyan. Apparently, the Jews during that era were lacking something, so that their redemptions were only temporary. Basically, as we are told explicitly in פרק ב' of Sefer Shofetim, the people sinned and had their land occupied; they then repented and liberated their land. This pattern kept on repeating itself with the people never repenting long enough to establish any permanence to their miraculous military victories. The message which this communicates to us is that if we want to usher in the final redemption, which Hashem promises will be similar to that of Yetzias Mitzrayim (עיין מיכה ז':ט"ו), we must repent in a permanent enough way that our redemption isn't undone a generation later. If we return completely to Hashem, we will be redeemed the same way that our ancestors were in Egypt, when the redemption was first beginning its last stages after forty years; if we don't, the redemption's effect will wear out after forty years, as happened with Devorah and Barak.
This message is certainly valuable, and hence we read about it in a Haftorah during the year, but it may be a bit too strong for Yom Tov, when we are not supposed to get depressed. Consequently, although we read Shiras Devorah as the Haftorah for Parshas Beshalach, we replace it with Shiras Dovid on the seventh day of Pesach. The most unique thing about Malchus Beis Dovid, the reign of the descendants of Dovid HaMelech, is, as the Rambam writes (פרק א' מהל' מלכים הלכה ט'), that although any Jew can potentially become a king, only Malchus Beis Dovid has permanence. As Dovid HaMelech himself emphasizes in his farewell words, "ברית עולם שם לי ערוכה בכל ושמורה," a phrase which indicates that his covenant is a permanent one (שמואל ב' כ"ג:ה'). Shiras Dovid is about how everyone from Dovid's own father-in-law (Shaul) to the Pelishtim tried to destroy him, but Hashem gave him the strength to "break copper bows" and to "jump over walls" so that he was able to survive. Shiras Dovid, unlike Shiras Devorah, ends with the description of Hashem as "ועשה חסד למשיחו לדוד ולזרעו עד עולם," emphasizing His eternal covenant with Dovid (שמואל ב' כ"ב:נ"א). This is the Haftorah that we like to hear as the Pesach celebration has us in good spirit.
The way in which Dovid introduces his song is by stressing how he always relied on Hashem; this is why Hashem saved him. If we want Hashem to fulfill His promise to us, each one of us must also, like Dovid, look to Hashem as "מגני וקרן ישעי משגבי ומנוסי משעי מחמס תשעני," meaning, as our protector upon whom we can rely to save us from trouble (שם כ"ב:ג'). If we can now truly return to Hashem, then the redemption which we have already begun to see in our own time will lead to a redemption that the Novi promises will not only match Yetzias Mitzrayim, but will dwarf it (עיין ירמיה כ"ג:ז'), rather than to the type of redemption that occurred in Devorah's time, which was only temporary.