Quiet means little noise. Quite means completely or very. A quiet room. Quite sure. Diction as a set is in word choice.
Silence versus very
If you can swap in silent, use quiet. If you can swap in very or completely, use quite. Quite please is the usual miss.
Later versus latter is another look-alike pair, in later versus latter. Then versus than is in then versus than. Grammar overall is in ACT grammar rules.
The bottom line
Little noise: quiet. Completely: quite.
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