Tradition separates the two. Some one who wants to write jazz will write from a jazz background relearning the basics in a new way. A jazz composer might begin to use extensions, sometimes nonsensically, where a classical composer wouldn't.
However, I think much of the division has nothing to do with composers. Composers have traditionally been fairly open minded, especially in the 20th century. Many "classical" composers where and are heavily influenced by jazz (e.g. Bernstein, Milhaud, Villa-Lobos [?], Satie, Ravel, Debussy [lots of cross pollination here, but seen more plainly in Ravel and Satie], Stravinsky, Zorn, etc). Similarly many jazz composers where/are heavily influenced by 'classical' music (e.g. Mingus, Herbie Hancock, Gil Evans, Dave Brubeck, John Coltrane, Jacques Loussier). The division is in the mind of the performers and the audiences. Many performers believe that they must learn to play in one style, but this is changing (eg. Wynton Marsalis).