The two words are synonymous when used to refer to a piece of art created in ancient Greece or Rome, such as the Parthenon and Venus de Milo. In addition, a classic or classical piece of art can be one created long after the ancient era, but inspired by the Greco-Roman culture or art, by depicting ancient deities or mythical events or following its techniques and methods of expression. Classical can refer to all forms of art, including sculpture, architecture and painting, as preserved in jars, but not music.
Classical music has no connection to the classical period, as there are no samples of what music used to be in ancient Greece or Rome. Instead, "classical" on this occasion describes western art music from the 11th century onward. Furthermore, classical music is also used when referring to the "classical period" of western art music, which flourished in western Europe from the death of German composer Johann Sebastian Bach in 1750, to the 1830s.
The word "classic" is also used to describe a high-quality piece of art. This use originates from the term's literal meaning "belonging to the highest rank or class" or "serving as the established model or standard," as Farlex Free Dictionary suggests. On this occasion, you cannot use classic and classical interchangeably, as this would alter the meaning of the sentence, from "an exceptional piece of art" to "an artistic expression inspired by ancient Greece or Rome."
As explained above, both classic and classical have two distinct meanings. Classic means both excellent and a piece of art from -- or inspired by -- ancient Greece or Rome, while classical is both a synonym to classic's second meaning and a period in western music. The only way to distinguish the meaning of the words in a given sentence is to use the context. For example, in the sentence "Oliver Twist is a classic novel," classic refers to quality, as neither the story was inspired by classical antiquity nor did novel writing exist back then.