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

Probably the phono stage, the preamp, and the power amp; in that order. Best of the are all pure tube. 

Just wanted to totally agree with @tannoy56.

Where it starts. Source. Pre.......power and all interconnections.

Of course each step could be the weak link in a system, but you would never know how good your speakers would/could sound if you don't give it a chance to be supplied with good input.

@willyht what has your experience been?

Since I started upgrading my system from mid-fi to hi-fi 6 years ago it has been 100% Audio Research. I have been through 9 different preamps and 4 different amps. I always saw the biggest impact from amps with the preamps offering more subtle differences.  The majority of these upgrades were with the same speakers.  I tried a few different speakers along the journey but always sold the contender and stuck with the original. 

I may be wrong but I have an impression that quite a number of audiophiles use different brands of preamp and power amp. Many also think that it is in fact more difficult to make a great preamp than power amp. Besides, synergy doesn't mean it must be one brand.