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

This year my answers have changed to this question. Previously I would have said the amplifier, but that was because I did not own a high end preamp.

Amplifiers matter up to a point that you need to drive speakers efficiently without straining the amp; not pushing it into distortion. To my mind, that is an amplifier that can deliver high current; not just watts per channel. This year I added a CODA S5.5 solid state amp, and an Aric Audio Transcend "Push Pull" EL34 amp.....both capable of high current. They truly transformed my system, even with a "mid-fi" preamp.

Then a couple of months later, I added an Aric Audio Motherlode XL balanced preamp.....and I learned what all of the fuss was about with a high end preamp. It took all of the magic of the amps, and improved on most everything in every way.

I also believe that the room conditioning is of equal importance, so much that I'm building a dedicated listening room, with proper diffusion and absorption, dedicated power with high end receptacles. Once that is complete, then I'll review what needs to be changed.

Yes, the preamp is the heart of the system.

But I have always felt that the “2 hands clapping” are the speakers and the amp.

The contribution of the preamp is more subtitle and surely subtitles are central.

But, IMO, the “broad strokes” belong to the amp and speakers.

I used to be in the anti-preamp camp for a long long time. I had evidence that preamps don't matter at all, they make zero difference. At least I couldn't hear it. Until one day, I got one and plugged it in. 

I am still kicking myself for selling a Rogue Rp7 pre w/ top of the line NOS tubes.

I have a well regarded SS integrated now but definetly lost a little magic.