I wouldn't say "totally" different at all. The main difference I hear is in the pick attack, the initial transient response. JK gets his smoothing at the front end where many/most of us going for an authentic Jerry tone get it at the back end. What I mean by that is that John goes for a bit of tube clipping at the front end of his preamp and lets the power amp and speaker end of things run clean and linear. By nature, that kind of clipping will be asymmetrical and contain a high proportion of even-ordered harmonics unique to asymmetrical clipping. Also, dynamically, it's a compressed sound, more akin to what we think of with bluesy or rock guitar tones. That compression may be why you hear Trey in the sound. What I hear with this approach is a very smooth, very sweet yet an un-dynamic sound. Very evened out. That compression happening so early in the chain kind of restricts the dynamic range of picking expression and volume control. In Furthur, the main difference I hear between John and Jerry is that John's pick attacks often disappear into the music, hidden beneath drum beats and Bob and Phil's attacks, and mostly we only get to hear the notes bloom after the beats and pick attacks. Jerry's pick attacks popped out with great presence and "pop" and clarity above the band. To me that's the big, glaring difference. But other than that, I find John's tone to be very Jerry, very clear and sweet, emotive, and serves the songs very nicely.
The approach that most of us here seem to take when going for a Jerry tone is to keep PLENTY of clean headroom thru the preamp stage to capture the full dynamic range of picking expression and then let the peaks and pops and transients get delivered thru with great force into the power amp and speakers and then let the smoothing happen there, at the power amp and at the speakers. It only works at high volume. That's when the JBL's do their own natural compression and subtle, sweet distortion. It's also when the power amplifier generates little amounts of peak limiting and subtle symmetrical clipping. It's that symmetrical clipping that, to me, sounds SO identifiably "Jerry". There's a clean-ness, a clarity, and a crisp, crackling focus in the clipping. That mechanism right there, the stressed power amp and speakers, I think, is central to getting that meaty, punchy, expressive Jerry tone where the pick attacks poke thru loud and clear and don't get buried under the band.
... and it's just like any other day that's ever been...