The JJ3(D) demo you speak of was made a little while ago by World Tree Games. It used the UT engine and looked quite nice. Excellent textures, models, etc. World Tree, once the game had reached pre-alpha stages, took the game to Epic, where it was recieved very well. But since epic is a developer, they needed a publisher, and at the time, they were with G.O.D. (Gathering of Developers), which had lost money on JJ2. Having done that made them reluctant to fork up any for JJ3, which they didn't--and the game died. That alpha still exists and has been circulating around the JJ community (as illegal as that may be, it's the truth).

It has :
No netplay *cries*
Several singleplayer "areas" including a forest-type-place, some mountiany-stuff, a town, a lake, and a castle, where the only boss is. There are some bugs in the game that make it difficult to get to the boss. It is, however, beatable. Though I forget what happens when you do.
Jazz doesn't have any of his special moves but he does have a "burst of speed" type thing where you can push forward twice in quick succession and get just that. (:
Guns "recharge" instead of having ammo amounts. So--the blaster recharges fast enough for you to fire continuously without problem, but the bouncer, say, recharges slowly so that you have to change weapons or stop firing.

Yeah...that's not all, by a long shot. See the JJ3 FAQ here : http://www.jazz2online.com/~j3f/

I've unintentionally mirrored much of its content here, but they've got more that is good.