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

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.

@inna +10!

Yeah, balance is the key. And speaking of tube equipment - tubes matter, a lot.

Both!!!

I’m not an engineer, but my uneducated guess is that the harder the speaker is to drive the more the amp matters in terms of sound quality.  My speakers are a relatively easy load and I hear bigger differences between preamps than amps, which would echo what others have said that “it depends.”  In the end, and no matter which one makes a bigger difference in a given system, balance is the key and that the amp and pre compliment each other in ultimately achieving the desired sonic characteristics.  That’s my take/experience anyway.

It is probably impossible to beat a great tube preamp.

Whichever, in the current setup, is less good.

The answer is always the same, it depends. That said it's hard to beat a great tubed preamp.

I am pretty sure this identical question has been asked in the past, and every time I have read it, the answers are (like almost all audio topics here) split with no clear consensus, and a lot of "both" answers.

I am going with "both," but the importance of each manifests itself in different ways. I’ve played around with a few different tube amps, and each amp change did something. I didn’t start to feel like my system was sounding truly "musical" until I upgraded from a digital HT preamp to my first tubed preamp. I upgraded that to a better tubed preamp and lost some warmth but gained air and detail (which to me means a better imaging and a better defined sound stage).