Why are people so meticulous regarding what they eat on Pesach? For example some people won't eat fish and/or fruit that don't have peels. However, what about meat? You don't see people saying, "Don't eat meat because it might be cheilev." And they are both kares-prohibitions (chometz and cheilev).
The basis of the custom is from the Arizal, who says that one who is careful from even a Mashehu (any amount) of Chametz on Pesach will not sin (accidentally) the whole year. Therefore as many stringencies as possible should be kept.
This is brought in halachic Achronim, most notably the Ba'er Hetev on Shulchan Aruch Orach Chaim (which is where it was most popularized).
So it is in pursuit of this promise/in recognition of the importance of it as personified by this promise that you see such stringency as compared to other Kares prohibitions.
Chametz on Pesach happens to be one of the only prohibitions even when it gets mixed with some other substance, doesn't become nullified (Batel Beshishim). The only other prohibition I could think of off the top of my head that has similar restrictions is Yayin Nesech (and other Avodah-Zarah related issurim).
Another difference would be the fact that Chametz is not always noticeably different than non-Chametz. Of course, There is an obvious difference between bread and Matzah, but some of the more nuanced forms of Chametz aren't as distinguishable (i.e. the difference between dough that rose for 18 minutes and dough that rose for 20). There is even a whole chapter in the Kitzur Shulchan Aruch (107) dealing with how to handle the flour in ways that won't become Chometz, and while I don't claim to be an expert about baking, I'm assuming the 'Chametz' that it's referring to is something slightly less noticeable than a bagel. A butcher, on the other hand, knows which piece of meat is Cheilev and which isn't, since it's easily possible to physically discern where it is. Even though I personally couldn't tell the difference between one piece of meat and another, I don't have to be concerned that it might be Cheilev, since I know that someone who DOES know what he's doing took care of it.
A relative of mine (no further source available) suggested that on Pesach, we became a new nation. Just as one is extremely careful with a newborn infant, for whom an injury or illness can be much more severe than for an adult, one is extremely careful with the laws of Pesach - the holiday of the nation's birth.