classic (adj)
1. of the highest quality; the most important of its kind
Have you read the classic novel War and Peace by Tolstoy?

2. elegant, but simple/traditional in style/design; not affected by changes in fashion
a classic navy suit
Some people drive only/collect classic cars (cars which are no longer made, but which are still popular).

3. very typical; something you would expect to find
You can see a classic (also classical) example of poor communication between parents and children in many families.

classical (adj)
1. widely accepted and used for a long time; traditional in style or idea
the classical theories of physics
classical and modern ballet
2. connected with or influenced by the culture of ancient Greece and Rome
classical literature/architecture
a classical scholar (an expert in Latin and Greek)
3. (of music) written in a Western musical tradition, usually using an established form (for example a symphony ) and not played on electronic instruments. Classical music is generally considered to be serious and to have a lasting value.
She plays classical music, not jazz.
a classical composer/violinist
4. very typical; something you would expect to find
These are classical (also classic) examples of food allergy.
5. (of a language) ancient in its form and no longer used in a spoken form
classical Arabic

the classics (n)
the most famous works of literature
I spent my childhood reading the classics.

classics (n)
the study of ancient Greek and Roman culture, especially their languages and literature
She studied/read classics at Oxford.

So, both classic and classical can be used for a piece of music but their meanings will not be the same.