Question

What is the musical element that distinguishes jazz music from most other styles of music?

Answer 1

Depends really. 
The question is tiny bit vague. But I could think of a few things. For one, it was one of the first music genres with beat. Which would become very important when Blues/Jazz slowly formed into Rock, Metal, and other more modern sounds. 
Also, Jazz uses classical elements, but with a pace. Classical music has pace, but it has no beat. Classical music can change pace and rhythm as desired without loosing too much quality. Blues/Jazz however, can't. It uses instruments and sounds all the same, but it has a beat, pace, and rhythm that cannot change without loosing control, or quality.  <span />

Answer 2

Answer:

The syncopated rhythms

Explanation:

Jazz frequently uses syncopated rhythm or rhythms on the "upbeat".

As far as method or sub-genre, I would say that improvisation is exclusively associated with jazz music.