This is confusing!
" Suddenly" is a synonym of "all of a sudden"
Nowadays " quite suddenly" means "rather suddenly", that is not as quickly, not as suddenly, not as unexpectedly, as "all of a sudden".
"All of a sudden" means "suddenly, unexpectedly"
Why nowadays ? Because "quite" has acquired the meaning of " rather" or "moderately". It first had this meaning in the 19th century and nowadays when someone says that something is 'quite '+ an adjective or adverb, that's usually what they mean.
"Quite" used to mean only "completely; entirely; fully; to the utmost extent". What's confusing is that it still does, sometimes. It does so in exclamations and when the tone of the speaker is emphatic; " It was quite horrid !"; "His behaviour was quite disgusting !" and " I found her piano playing quite delightful"