The reason why they are killed is that כל השוכח דבר אחד מלמודו1 מתחייב בנפשו. But what is למודו that if one forgets it he should be killed? Well we can figure this out from the laws of Purim. In Shulchan Aruch O.C. 695:2 R. Moses Isserles writes:
וי"א דא"צ להשתכר כל כך אלא שישתה יותר מלימודו
And some say that he does not have to get so intoxicated; rather he should drink more than he usually does.
Thus, לימודו apparently refers to what one usually consumes. In the case in Pesachim, the thing they have forgotten is precisely which animal they normally consume; thus, they get the death penalty.
As for why the Mishnah needs to tell us that they are exempt from Pesach Sheini, we need to understand how the Jewish court system worked back then. The Mishnah in Makkot 1:10 states:
סנהדרין ההורגת אחד בשבוע נקראת חובלנית רבי אלעזר בן עזריה אומר אחד לשבעים שנה רבי טרפון ורבי עקיבא אומרים אילו היינו בסנהדרין לא נהרג אדם מעולם
A court that kills once in seven years is called "destructive". R. Elazar Ben Azariah says: "once in seventy years". R. Tarfon and R. Akiva say: "If we were on the court no man would ever be killed".
Thus, it is almost certain that one who earns the death penalty around Pesach time will not have been executed a month later. He would only be executed if it was at least seven or seventy years since the last execution, and according to some opinions even then he would not be executed. Thus, the Mishnah (in Pesachim) teaches us that in the likely circumstance that they have not yet been executed by Pesach Sheini, they are exempt from Pesach Sheini.
1. As far as I know none of the Sages' formulations of this idea use precisely this word, but some acharonim paraphrasing imprecisely do use this word so it must be a close enough match.
