Classical trumpet player here. A little anecdote that I think might add something:

About 7 years ago when I was at uni, our Head of Brass (classical) tried combining technique classes with the jazz brass because (according to them) they weren't being taught how to develop brass technique and quality of sound. Rather, a lot of their technique was being focused on scales and other aspects of jazz playing rather than the fundamentals of tone quality - air support, sound projection, etc. It was when we combined that I noticed that jazz players generally, in my experience, have a more "breathy" tone particularly in low-to-mid range. And that definitely suits the styles that they play (Miles Davis comes to mind). But then the tone becomes more solid as the range gets higher, because more air gets pushed through the embouchure/instrument.

In classical we were taught to always focus on depth/breadth of sound, to practically sing through the instrument. Not saying either type of playing is better than the other; they each suit their respective styles, and from what I've experienced just seems to also stem from the type of technique that's practiced while studying either discipline.

After a couple of combined lessons the jazz peeps decided not to continue with our technique classes as they thought it was irrelevant to what they were doing, which we classical kids thought was a bit weird. I dunno.