Why do amps sound different?

A very likely reason two amps can sound different, yet measure similarly on the test bench, is that the measurements typically taken are not done under the real load an amplifier will encounter with a dynamic waveform. Secondly, I'm not sure the typical bench measurements are measuring the right things regarding how an amplifier will sound. For a simple example, what if THD is basically the same, but the distribution of even and odd harmonic distortion is different among the various amps. Or what if various parameters, alone perhaps not significant, combine in ways that alter our brain's interpretation of the sound?

I think it's a not so simple a question you've asked. In my opinion, the best amplifier manufacturers have a basic measurement model they've developed/trust that allows them to accept or reject a given design before putting much additional time and effort in developing it. I would also guess that any manufacturer that claims to develop a design soley from ear, beginning to end, is probably not being truthful, spent a lifetime getting one design "right", or will not have much consistency in future designs. My point is that an audio system is a complex system, much like the weather. We can know quite a bit about the "averages", but it takes a very complex model, with alot of measured inputs, to have a reasonable chance of predicting the weather with any degree of accuracy.

TIA - I am most certainly not any kind of expert, but I did sleep at a Holiday Inn Express (once...)

For starters, if they do sound different then they do not measure exactly the same - period. You may be referring to a single aggregate measure (amps, etc.) when you say that they're the same but if they sound different then the detailed measurements really are different. All amplifiers vary from the ideal of "a straight wire with gain" - it's in exactly how and where they vary that leads to perceived differences.

For one reasonably detailed explanation you could try this:
http://sound.westhost.com/amp-sound.htm

I would bet that they don't really measure the same on the test bench as you say, if they do the tests are not comprehensive enough. Specs can't be looked at individually. When people say that two amps measure the same, they often look only at WPC and THD and, for the very advanced, s/n ratio.

I am far from an expert, but from what I have read, measurements aren't always a barometer of sound. As a subscriber to Stereophile, many times the measurements indicated on some high priced amps are not really up to snuff, yet despite the measurements, the reviewer will swear on a stack of bibles that the amplifier sounded excellent. I think every company has it's own unique sound bias and will seek to design it in their product. The quality of parts and even the type of wiring (copper vs silver) will also determine the sound quality. For instance, even with the same measurments, tube amps almost always present a different sound quality than solid state.