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

@inna +10!

It really depends on how neutral and transparent each piece in question is. In a top notch system, the weakest link will stick out like a sore thumb...not so much on a system with multiple veiling and smearing issues. Plus, as mentioned, the amp/speaker interface matters.

At some level everything matters.

Each are important but a high quality, low noise high gain preamp can bring your system to a higher level of excitement and dynamics than a basic preamp connected to a powerhouse amp.  

I have been pondering this for a while but don’t have a succinct answer yet.. but what I have goes like this.

Yes, both are important but the preamp is the heart of the system and as such is more important. However, I always remember the huge effects on bass presentation and gross change in the rest of the frequencies given by the amp.

I think it is that the preamp sets all the nuances across the spectrum… it affects much of the personality. Then the amp makes big accentuations and attenuations on top of that. While the amp makes a few big changes the preamp makes more nuanced changes across the board which add up to being more important. When you are talking similar quality. I tend to error in spending more on a preamp than amp. 

Or maybe this, if you don’t get the preamp right then the amp can’t get it right.

Both can, and do affect sound quality. Every component and interconnection in the chain affects the quality. Depending on your source you don't always need a preamp in the system.