How is "Gypsy Jazz" different from regular jazz?

Gypsy Jazz is just one specific style of jazz. You could think of different styles of Jazz as the same as a language with different dialects. It’s much of the same vocabulary as so called “regular jazz” but it has certain stylistic characteristics which give it it’s unique sound.

In the case of Gypsy Jazz, it was most associated with Django Rheinhardt and Stephane Grappelli who defined the sound in 1934 in Paris. The best way to understand the music is to listen. You can get a good sense of what makes the sound unique by watching this documentary on how the music got started. It was a marriage of the violin music of French-Italian Stephane Grappelli and Belgian-born Romani-French acoustic guitarist Django Rheinhardt. It is an older style jazz which brought a lot of attention to the guitar and violin as solo instruments for the first time in jazz.