There is no more marked area of difference between classically trained musicians and musicians trained in jazz than mastery of rhythm. One big difference is the improvisational aspect of jazz. While there are widespread jazz compositions, in more general terms, the pieces are vehicles of exploration. That's significant if you come from the classic side, where everything is written down.
For decades, composers of jazz and classical music have looked the other side of the fence. Classical composers envy the melodic enthusiasm, spontaneity and open emotion of improvisation; jazz musicians focus on the major scale, on the colourful and rhythmic flexibility, and on the respect given to classical music. Classical music is usually performed by an orchestra, and each musician plays a specific part. And the focus is on melody and harmony rather than improvisation.
Jazz is influenced by classical music, but also by blues and spiritual music, which gives it its unique sound. I have been a classical musician all my life, I have been composing classical music and playing it on the piano, but lately I want to make a transition to jazz and I am curious to know what are the main differences between classical and jazz music when it comes to composing. Nor do I know if there is any historical revisionism; for example, Mel Tormé was considered a jazz singer in his time, but I wonder if contemporary jazz artists (who are always instrumental) would have the same opinion, that is, primitive jazz music was played in small ensembles that used clarinet, tuba, cornet, baritone, drums and piano. We see this a lot in many other forums: a lot of people with a solid background in something other than jazz want to learn jazz.
Few jazz musicians considered pop melodies from the 1950s, 60s and beyond worthy of attention (with notable exceptions), no doubt because they didn't want to be popular (as jazz had been before World War II).