New generations of hardware can allow for new experiences beyond anything possible on the system being replaced; whether in terms of graphics, gameplay, functionality, audio, etc.

Which leap from a specific system to its successor do you think was the most impressive, and why?

SNES to N64. While the N64's lack of RPGs frustrated me, Super Mario 64 and OOT/MM were very impressive. Every generation subsequent to that seemed to become more iterative than revolutionary, though they all had their gems.

SNES to N64
Saturn to Dreamcast
Genesis to Saturn

2D to 3D is the biggest leap ever. DC wasn't just powerful at launch. But features ahead of it's time.

SNES to N64,

NES to SNES.
SNES to N64.
PS1 to PS2.

FM Towns Marty to FM Towns Marty 2

2D to 3D is the obvious pick here.

But going by games, which is the more interesting topic IMO, Super Metroid, Witcher 3 come to mind.

PS1 to PS2
Saturn to Dreamcast

2D to 3D was the biggest leap IMO and Nintendo did it best, so SNES to N64 for me. We are now at a point where the leaps are smaller every generation. 

SNES to N64 for sure.