Which has a greater influence on sound quality, the amp or the preamp?

I think it depends on the equipment used so Both.

I've gotten the most noticable differences in system sound based on (descending order of magnitude)

Speakers/Headphones 

Room (for stereo listening)

DAC/Cartridge 

Subwoofer placement/integration 

Phono stage

Amplifier (tube types)

Preamplifier (with/without tube type)

Streamer

I haven't done DSP but do have a DIRAC unit collecting dust and based on what others say it could fall somewhere between subwoofer integration and amplifier.

Yes to both unless have a mid-fi or worse amp that will mask the change a preamp brings. 

Which is more important to walking? The left or the right leg? 

Not a good analogy.

But as,someone stated a good tube preamp does wonders to enhance the SS amp.