The blood on the doorposts represents the life of the lamb given to redeem those within each participating house (see Exodus 13:13-16; 34:19-20), not to symbolize forgiveness. God does not draw attention to the Israelites' sins in His Passover instructions, even though the Israelites were sinning - grievously, in fact. Through Ezekiel, God says that the Israelites as a whole were unabashed idolaters at this time, and God nearly destroyed them, then and there:
The Israelites not only had idols in Egypt, as this says, but other passages show that they were still carrying them in the wilderness. This means that some Israelites, maybe many of them, had idols in their homes even as they kept the Passover. That may be shocking, but what it shows is, first, God's incredible mercy, and second, that the Passover was not about cleansing Israel from sin, which requires repentance. The Passover had another purpose.
Now, why would God destroy one nation of idolaters (that is, Egypt) and yet deliver another nation of idolaters (that is, Israel)? God's displeasure in Ezekiel is obvious, and He gives no hint that the blood of the Passover lamb was for atonement. Israel's sins weren't being dealt with - God was overlooking them. This is why, according to Strong's, the Hebrew word for Passover (pesach) means, a pretermission; [that is, an] exemption. The word pretermission is hardly used today, but it basically means, an omission. To pretermit means to let pass without mention or notice. It is similar to mercy or grace (see Numbers 14:19; Romans 9:15-18). When Israel was in Egypt, God passed over them and their sins. Moses says this in Exodus 12:27:
The first difference, as mentioned, is that the emphasis of the Passover is on eating. In contrast, the whole purpose of a sin offering is to show atonement or propitiation, which did not happen in Egypt. God prescribed the sin offering for unintentional sins and admitted guilt (Leviticus 4:2, 13, 22, 27), but the Israelites kept the Passover in Egypt while refusing to get rid of their idols (Ezekiel 20:5-9; see also Exodus 6:2-9; Leviticus 18:3; Deuteronomy 9:7; Joshua 24:14; Amos 5:25-26; Acts 7:42-43).
