What is the difference between "jazz music" and Frank Sinatra?

Back in the day Big Bands like Tommy Dorsey's band and such also employed singers to sing some of the songs they did. Some of these singers became quite famous singing and soon songs were written around the singers and the band took more of a back seat. A vehicle for the singer cause that singer would make the band leader a bunch of money. So yes to answer your question Frank Sinatra did sing jazz. It's kind of like what came first the chicken or the egg. In this case the music came first then came the singers.

Jazz is a style of music, Frank was a person. ... I'd put Frank and company in the category of "crooners." Male singers of popular standards. http://en.wikipedia.org/wiki/Crooner (hey look, there's Frank's picture right on the page.)

One definition of jazz is that it has to "swing", and there has to be improvisation. I think of Miles Davis, John Coletrane, etc. to be jazz in this sense. I think Frank Sinatra is more like Swing or Big Band. As far as I know, there's not much improvisation in his music.

Sinatra and the people you mentioned are stylist.

In my opinion, Frank Sinatra's music is definitely more like Big Band. You see, like many other genres in music, sometimes they can be more "instrumental"/ non vocal. If you listen to more instrumental jazz, like for example, Bill Evans [just pulling one out randomly], you'll notice it's more improvised jazz, in his case, with the piano, since he was a pianist and it doesn't necessarily have any vocals in it. Now, Frank Sinatra on the other hand, as well as Tony Bennett, Dean Martin, etc were more vocal and had much bigger instruments added to their bands, many more melodies, rhythms and heck, they were even louder! which is why the term "big band". Hope it helped...