Question 1: How is the Pesach rule of על השובע different than the general Kodshim rule of על השובע.
Although the rule is best known in the context of Korban Pesach (Rambam 5 Korban Pesach 3), an identical rule applies to all korbanos (Rambam 10 Maaseh Hakorbanos 11, from Temura 23a). See Zevachim 30b in middle of the daf and Rashi that brings the Toras Kohanim. So the question is, why do we need a drasha specific to Korban Pesach. And having a drasha specifically by the Korban Pesach, how is this din על השובע different from the general kodshim rule of על השובע? (Rashi, Pesachim 83a, says that there is no special din by Pesach, and that the famous על השובע by Pesach is just an application of the general על השובע of all Kodshim. But most Rishonim do not hold like Rashi, like the Rambam in Mitzvos Asei #56 who quotes the Mechilta in support of his pshat.)
So the Brisker Rov answers that it appears in the Rambam that the general rule of על השובע only applies to Kodshei Kodoshim, but not to kodshim kalim like Korban Pesach. The reason for this distinction is that the mitzva to eat kodshim only exists for kodshei kodoshim, while the mitzva for kodoshim kalim is only to see to it that the kodshim are eaten, but not necessarily that you eat it yourself (as the Beis Halevi says, though he says it regarding both Kodshei Kodoshim and Kalim.) The chiddush of the passuk here is that even though Korban Pesach is Kodshim Kalim, there's a din of al hasova anyway. The Rov says this might be because of the unique mitzva of eating korban pesach; the purpose of the korban is that it be eaten, unlike other korbanos where the purpose is the sacrificial service and the eating is secondary. Here, the eating is primary.
Elsewhere, (in the Grach al Hashas page 21 to Pesachim 41a) he says that the general din of על השובע by kodshim only applies to Kohanim, for whom there's a din of Lemashcha, but it does not apply to non-Kohanim that eat kodshim, like Shlamim or Todah. So the chiddush by Pesach is that it applies even to non-kohanim.
With some thought, I think we ought to be able to think of more answers. For example:
1. The drasha in Torah Kohanim, as brought in Temura, says that if one has an abundance of korban meat to eat, the meal should consist entirely of the korbanos. But if there is little meat to go around, one should eat other foods to take the edge off his hunger and only then eat the korban. We see that the din of al hasova by kodshim does not mean that you have to eat every kezayis al hasova. It only means that you should ensure that by the time you get the finish the meal, the last kezayis will be al hasova.
2. The Chinuch, based on the Yerushalmi (Pesachim 6:4, brought in Tosfos Pesachim 70a D'H Lahv) says that the reason the Torah prohibits breaking bones of the Pesach is so that people should not eat it ravenously, breaking bones and sucking out marrow. They should eat it calmly. The Chinuch is reading this rule in consonance with the din of Al Hasova. If the general kodshim rule of al hasova were all that mattered here, that would mean that you could begin eating the Korban Pesach with a strong hunger, and the Chinuch's idea wouldn't make any sense. Evidently, the Chinuch assumed that by the Korban Pesach, you need to eat every single kezayis al hasova.
So we already have one answer. It could be that by other kodshim, the din al hasova only means that you should end your achilas kodshim with a kezayis of some korban al hasova. It's a din in the participants in the meal, not in the kodshim themselves. But by Pesach, the din of al hasova applies to the korban itself, and requires that every single bite be al hasova. Unfortunately, the Rambam in 8 Chametz Umatza 7 and 9 indicates that he holds that you eat the Pesach during the meal and then again at the end of the meal, just like other kodshim.
PART II, addressing Question 2. As it turns out, Question II leads us to an answer on Question I.
There is something very strange about the basic concept of al hasova. It seems to be diametrically opposite of the general rule of Mitzvos. That is, the Gemara in Pesachim says that one should not eat a lot before Pesach, in order that he should eat Matza "Le'tei'avon," with desire, with a strong appetite. If by kodshim the mitzva is to eat it al hasova, why is the rule by matza exactly the opposite? Why by one is the mitzva hunger, and by the other the mitzva is enjoyment davka without hunger?
We have a rule of מצוות לאו ליהנות ניתנו. The logic of this rule is that any pleasure derived from fulfillment of a mitzva is viewed as tertiary and legally irrelevant; the only result of the act that is recognized as significant by the halacha is the kiyum hamitzva. On the other hand, you have the Taz in YD 221:43, and the Rabbeinu Avraham min Hahar in Nedarim 48, that say that because the whole point of limud hatorah is to learn with pleasure, the rule of lav lei'hanos doesn't apply. The Rashba there, and the Ran there on 37a, disagree. Rav Shteinman Shlitah in his Ayeles Hashachar on Nedarim 16b starting from "Be'omeir" and especially on the next page DH Ve'Efshar) says that it according to some rishonim, even if we would agree with Rabbeinu Avraham that the purpose of limud is to have hana'a, that is all the more reason to say lav lei'hanos nitnu, because the pleasure is the mitzva, and the pleasure we personally derive is considered secondary to the purpose of serving Hashem. (see, e.g., שערי יושר א:ז ד"ה ובדרך, קובץ שמועות חולין לט, but see farkert (c.f.) רשב"א סוכה לא: who says pshat in lav leihanos because hana'a ruchnis doesn't matter. This machlokes is discussed in the פרי חדש או"ח תקפו-ה and the Sefer Hamakneh 22:12. For more mekoros than you can shake a stick at, see Sdei Chemed volume 4 page 205.)
So it turns out that there are mitzvos where the whole kiyum of the mitzva is to enjoy the act. By such mitzvos, the more hana'ah, the more mitzva, and so we eat le'tei'avon. This is true where the mitzva is the achila. But even though the hana'ah is the mitzva, we still say Lav leihanos nitnu, that because we are motivated by the chiyuv, not the pleasure, and any pleasure is trivial and halachicly irrelevant. So we do them in an ofen of hana'a in order to be me'kayeim the mitzvah with hiddur. In other mitzvos, though, where the kiyum is not the pleasure, then kal vachomer the hana'a is irrelevant. By those mitzvos, since the hana'a is not the tachlis of the mitzva, the hiddur in the avoda would be to demonstrate that the hana'a is not important, and so it is best to do them "al hasova." This is true by achilas kodshim, where the mitzva is not to eat it, but rather to "see to it that it is eaten."
Korban Pesach is a strange kind of mitzva. On the one hand, it is kodshim, where the best thing to do is to eat it al hasova. On the other hand, it's a mitzva to eat it, like any achila shel mitzva, and is such cases, the best thing to do is to eat it le'tei'avon. So what do you do? The Torah is mechadeish that even though it has aspects of other mitzvos, the kodshim aspect is dominant, and therefore it should be eaten al hasova. And even so, the matza that accompanies it, the Al matzos umerorim of the Korban, should be eaten le'tei'avon. (See Sfas Emes below.)
This explains why some mitzvos require sova and some require tei'avon. It also explains why the Torah needs to tell us Sova by Pesach even though it would be covered by the general din sova of kodshim anyway.
(My shiur answered the Le'tei'avon/Al Hasova question differently: that Al Hasova applies only when you're eating Mishulchan gavo'ah. Once you say that Al Hasova is only for korbanos that are eaten Mishulchan Gavo'ah, you can say like the Brisker Rov above in explaining why you need a new halacha in Korban Pesach.)
Also: in the previous post, I quoted a Sfas Emes on the topic of the difference between the Matza of Ba'erev and the Matza of Al Matzos. This Sfas Emes happens to be equally relevant to this discussion of Sova and Tei'avon, so I'm quoting it again here.
I do want to mention though, that his statement that the Matza and Maror of the Korban is not to be eaten le'tei'avon, but rather al Hasova, seems to be contradicted by the Mechilta brought in the Kesef Mishna in 8 Hilchos Korban Pesach 3, where the Rambam says מצוה מן המובחר לאכול בשר הפסח אכילת שובע., the Kesef Mishna says: