The generational jump from the primarily 2D 4th gen consoles to the 5th seems to be widely considered the biggest generational leap, but what of those that followed? 

In the 3D era (5th gen onwards) what do you think was the biggest leap from one system to its successor?

No doubt: PS2-GC era coming from PS1-64. There's no discussion here, as, to be very blunt and honest, following generations have been pretty much iterations of that early 2000's that we have been experiencing ever since.

Edit: thanks for noticing, Curl! So, I was writing while tired and misread the post! I thought we were talking generations. But then it's even easier: PS1 to PS2. I couldn't bear PS1's 3D games and pretty much can't to this day (I truly only have fond memories of Symphony from that console). 64 was way better in almost every way.

Which specific system to the next? N64 to Gamecube? PS1 to PS2? Saturn to Dreamcast?

N64 to Dreamcast. Or if we mean from one company ps1 to ps2. Dont really think theres much room for debate. 

5th gen to 6th gen. 

Many might say PS1 to PS2 but to me it was Dreamcast. Shenmue blew anything out of the water in 1999 (. Not just console but even PC and arcades. Soul Calibur looked miles better on DC than arcade and much better than Soul Edge. Sonic Adventure whale chase was the Next Gen has arrived moment for the media. Of course a few years later people were taken back by MGS at E3 and by that point DC was looking a bit old already.But in 1999 my god. I followed Shenmue in Magazines since 1998 as Project Berkley. The idea for me of picking up random objects in a room. Messing with light switches. I needed it. That blew my mind. The world was so detailed and character models were so good EGM did a piece telling people no these are not pre-rendered but in game models.  Same with Sonic Adventure. The weather effects and high rez textures. It looked as real as real gets to me back then. I have never been blown away like I have with Dreamcast ever since. PS2,Xbox,Gamecube would go on and have better looking games in the end of course as they has more capability and a longer lifespan.

360 Kiosks showed games that were not that amazing to look at. Kameo which looked like the Gamecube/Xbox build from Magazines. CoD II which was whatever. PD has a horribly ugly art style that again just looked like a reg Xbox game. Took until Gears and Dead Rising to finally show Next Gen is here.

Oh, I'm tired here and misread your post! I thought we were talking generations. I'll correct my post. 

Yeah, 5th gen to 6th gen was certainly the biggest perceivable leap because we saw the most noticeable improvements but since you want a specific system and its direct successor I'll go with N64 to Gamecube simply because of the added cartridge to disc data difference. Going back to even the best looking N64 games like Banjo still looked pretty dated graphically after playing Star Fox Adventures. Rogue Squadron to Rogue Leader, F-Zero X to GX, Smash 64 to Melee - so much 60fps goodness on the Cube with such a massive leap in texture quality.