"the knife" is direct - we don't need a context to explain it. The speaker was supposed to wash and particular knife, forgot to do so, and is now apologizing for the failure. It's a compact and rather clear set of facts, with obvious implied facts.
"a knife" implies that there is some unspoken need for a clean knife, and the context - or lack of it - doesn't give any idea as to what that use is. The implication is a large set of facts which exist outside the sentence, a set of facts which is not at all clear. That makes the sentence detached from whatever reality those facts present. In short, too many unknowns are created by using "a knife".
They're quite close but not exactly. Generally, it's more or less like "sorry" vs "excuse me" - 'прости' expresses more regret than 'извини'.
One uses 'прости' when feels guilty and regrets doing something (even if the action wasn't intentional) while 'извини' is used when your actions were accidental and not intentional or you couldn't prevent / foresee something from happening or you don't regret doing. One also uses 'извините' to draw someone's attention, like "извините, не подскажете, как пройти ..." / "Excuse me, how can I get to ..., please".
These are just broad / rough definitions combined with subjective perception. The words are very often used interchangeable or the usage could be affected by regional / family traditions, habits, one's society circle etc