As nouns, there is no difference. "Quotation" is ever so slightly more formal.

Old English is synthetical and strict in morphology, as like in Russian or older Russian; but it has become more and more analytical （of the type of Chinese today) later on. As an example, quite a lot of verbs have been used as nouns withought morphological (formal ) change, or even dispite the existing nounal forms for them. The new usage is often colloquial, or in some special fields( IT, for instance). Quote used as noun is just the case.

"To quote" is the verb that means both "repeat something, cite" and "set a price to" (Цитировать/котировать) a quotation is the substantive of that verb (цитата/котировка) But a quote used as a substantive itself can also mean "a part of somesthing", mostly used in financial contests. They come from latin, and have the same use in italian (to quote= quotare, a quotation = una quotazione, a quote= una quota)