Quite =довольно (often with positive ideas)
Ex. I’m surprised you haven’t heard of her. She is quite famous.
It’s quite cold. You had better wear your coat .
Lucy lives quite near me, so we see each other quite often.
Quite goes before a/an:
Ex. quite a nice day; quite an old house .
Sometimes we use quite + noun(without adjective)
I didn’t expect to see them. It was quite a surprise .
We also use quite with some verbs, especially like and enjoy;
Ex. I quite like tennis but it's not my favorite sport.
Quite sometimes means completely = I quite agree with ya. I'm quite sure.
Not quite =not completely
Rather = довольно таки (with neagative words and ideas)
EX. It's rather cold, you had better wear your coat.
Let’s get a taxi. It's rather a long way to walk.
She quite intelligent but rather lazy.
When we use rather with positive words(nice/interesting etc.) it means unusually or surprisingly for ex. rather nice =unusually nice/surprisingly nice/nicer than expected;
EX. These oranges are rather nice. Where did you get them?
Rather can go before or after a/an so you can say;
EX. a rather interesting book or rather an interesting book.