Roman public religion held that a rite performed incorrectly had not been performed. The category of fault was wide: a word misspoken or omitted, a gesture in the wrong order, a defect found in the victim after it was opened, a noise or an interruption at the wrong moment, an ill-omened event during the ceremony. Where the fault was discovered the response was not to apologise but to begin again, repeating the whole ceremony from the start, and to offer a piaculum for the error itself. Public festivals were repeated on this ground, sometimes more than once, which is expensive and was accepted as the cost of doing the thing properly.
This is a different logic from the one that governs the offerings of the Israelite system, where the classes of offering are matched to classes of wrong committed by a person. The Roman fault is procedural. Nothing has been done to anybody; a form has been broken, and the piaculum repairs the form. It is also a different logic from the vow, which promises a future offering in exchange for a future benefit and creates an obligation running forward. A piaculum settles an account rather than opening one, and expects nothing in return.
The offering itself was usually modest, commonly a pig, and its size bore no relation to the gravity of the occasion, because what was owed was an acknowledgement rather than a compensation. That the same word covers both the fault and the offering that repairs it is a piece of Latin that tells the reader where the emphasis lay.