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

They both offer different sound quality and sound characteristics.  

"Is it the inside dirt or the outside dirt that matters most?" assumes a nonexistent hierarchy.  It's the side of the glass with the most dirt tempered by whether fingerprints or water spots bother you more tempered by what floor you are on and how tall is your ladder.

The CLOSEST you get to a LIVE performance with a Wide and Deep soundstage is with a Tube preamp and a Solid State amp. Period.

@mbmi I get all of that using my passive (battery powered for pitch black backgrounds) with a tube phono stage and SS amps. 

If your preamp or source is lacking, it will not matter how good your amps and speakers are- they cannot make up for signal loss or coloration! The preamp has to be right!

Amplifiers tend to have the most distortion of any of the links in the electronics chain (loudspeakers tend to have even more). Because of this its important than the distortion of the amplifier be benign (lower ordered harmonics) since the "sonic signature" of any amplifier is the distortion it makes (IOW you hear distortion as tonality).