It can't simply be that the word "quote" seems more formal than "saying" can it?
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For what it's worth, "quote" and "quotation" are not strictly synonyms. Although interchangeable in popular use, "quotation" is a noun and "quote" is a verb.1 The shortening to "quote" has become more common and (dare I say it) accepted, but for the nitpicky among us "quotation" is still the only acceptable noun.
In response to your question: a saying is generally pithier, concerned with a single kernel of wisdom, and is not necessarily attributable to a single person, piece of literature, or entity.2 A quotation, in comparison, is almost always attributed and is not expressly focused on a kernel of wisdom.