Quite is an adverb and means entirely, as in the highest degree or to the fullest extent (It’s not quite as bad as all that!), or rather as in ‘to a considerable or great degree’ (It’s quite disgusting to think of living off eating grub worms!) and is often said in agreement with the subject under discussion (Don’t you think he needs to wash up before fixing our food? Quite!), or nearly, most often used with a negative to indicate that something has almost reached a state or condition and which may indicate an indefinite time frame (The dress is not quite finished.) Quite emphasizes exceptional quality, indicating something to be remarkably good, fine, attractive or otherwise admirable or impressive.
Quiet has so many meanings that I’m listing them each with an example within parentheses just following the meaning. Not noisy (in the quiet of the forest), still (in a quiet corner of the room), done in private (I’d like a quiet word with you.), undisturbed (a quiet life away from publicity or trouble or disturbances), relaxing (a quiet evening at home), not showy (a quiet wedding instead of the grand, showy, pretentious thing her mother wanted), restrained (the doctor’s quiet manner), unspoken (not expressed in words, as in a sense of quiet optimism), not busy (in the bad economy business is a little too quiet), calm or motionless (a quiet sea), to become calm and quiet or make somebody calm and quiet (He sang lullabies to quiet the baby. Will you all just quiet down please?), to allay anxiety (He spoke softly while quieting her doubts.), on the quiet (done secretly; Give the widow this money strictly on the quiet so she won’t know it came from me.).
Nearly all of these can be summed up as being low in audible volume, acting in a calm manner, or some combination of the two.
Dictionary.com has quite a cute quote I’m sharing with you today. "Better is a dry morsel with quiet than a house full of feasting with strife." -unknown author
Because of the diversity of the meanings of both words, the only hook I came up with to aid your memory is a bit absurd, but may help. When you want things to be quiet or to be done quietly, you want to suppress public notice or to keep it silent. The word silent has two syllables, and so does the word quiet. Quite most often may be substituted with the word yet or yes, and all three of them have only one syllable. Inane hint? Quite!