Why is the "the" the only option here? In my head i imagined a situation where i washed some cutlery and now someone needs a knife, so i say: "Sorry, i forgot to wash a knife".......what would be the correct russian to use in this distinct case ?
"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
I guess nobody answered your question which is exactly like mine. Anyways, I would have translated "прости" as "forgive me, I forgot to wash the knife" but Duolingo considered my translation wrong without any explanation, as usual.
In English, they are pretty much interchangeable and so I think either should be accepted. Are they used for different purposes in Russian?
I wrote "Мне жаль, я забыл помыть нож." Is the use of "Мне жаль" wrong in this sentence?
"Mне жаль" is not normally used in situations like this. Translating "I am sorry" as "мне жаль" is more suitable when someone is offering their condolences, rather than for the apology.
"I am sorry i forget to wash the knife" in my personal opinion, that sentense sounds strange
In my variety of English, "wash up" is better than "clean" here, and "clean" is better than "wash".