In a sense, jazz is the branch between the Romantic period (late classical music in the 1800's) and Rock and modern music. Jazz has much more freedom and emphasis on expression, while classical music was reserved and structured.
Of course this is oversimplified.
Some jazz characteristics are the emphasis on improvisation, freedom of expression (it's hard to play "wrong" notes in jazz), and a strong attachment to emotion.
Classical music has charactersics of structured, many rules, and controlled beauty.