Hi, Nathan. Hi Magin. Nathan's explanation is detailed and perfect, but I'm afraid Magin may be still confused. I'd like to give some understanding of mine, from the perspective of a Chinese-speaking English learner. If any mistakes, please tell me if only you would like.
There are corresponding Chinese words for each of the couple (adjective) words. Classic: 经典的, and classical: 古典的. Knowing this will make a good grasp of the relation of these two English words.
1.classic ≠ classical : 经典的,优秀的 used for modern things, or without meaning of eras or times
2. classic = classical: 典型的,具有典型特征的 for modern...or without ...times
These two uses will not make confusion, I think.
3. classic ≈ classical:古典风格的,老式的,典雅的, 传统的 used
(a) for modern (and often concrete) things: suit, design, furniture
(b) for good things that exsist but are not made any more: cars, furniture
classical: 古典的,传统的, 经典的 used for widely known and accepted (at least in the past) things, often abstract, that have a pretty long history and enjoy high prestige: ballet, theories
When classical is used for music (正统的,传统的,not rap, rock and roll) and language (古文的,文言的),it's fairly easy to understand.
classic/classical music of 1950s: 典型的/优秀的1950年代音乐
classical music of Russia: 俄罗斯的传统/经典音乐
a piece of classic music: 一部优秀的音乐作品
classical Chinese:文言文,古代汉语
So you can see, in Chinese we sometimes exchangeably use 经典的, 古典的, 传统的 too. With most of these uses, we do not have trouble in accurately understanding the users. It's the same about classic and classical for the English speakers. But, by contrast, I think they may have big trouble in making certain the relations between the few Chinese words.