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

Now that I've gotten my morning snark out of the way, my answer is that they are both critical but they perform different functions and thus their relative importance is contextually dependent. 

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.