Mark, as Ian and Chris point out, what you were referring to is termed "slew rate". This is the rate at which the amplifier changes voltage level over time. The usual parameter is given in volts per micro second. When a frequency during music appears which is higher(faster)than the previous frequency or is louder, the amplifier has to supply more voltage because of the higher level and do it more quickly because of the higher frequency. This is needed to produce that note at the required level and with low distortion. Note that this has nothing to do with sharper transient response or a perceived "tighter, quicker" sound; it's just doing the frequency accurately with low distortion.
The required slew rate at any instant in time is given by 2 x pi x freq. x voltage(peak). For example, to reproduce 20,000Hz at 100 watts(i.e. 40V peak, 28V RMS for 100 watts into 8 ohms, since power equals voltage(RMS)squared/impedance)the slew rate would be about 6.28 x 20,000 x 40 = 5,240,000(in micro-volts)or a slew rate of 5.24 volts per second.
This number contrasts with numbers on the order of 50, 100 or even more which some amplifiers proclaim, and is unrealistically high as is, because no music recordings require full power at 20,000Hz. Many years ago Baxandall
tested LPs for maximum slew rate required and found that 0.5 volt per second was all that was needed. British amplifier designer Michael Renardson updated this work for CDs, as shown here
and found a maximum requirement for 100 watts into 8 ohms of 2.5 volts per second(would be 5 volts per second for 400 watts). All receivers/amplifiers of a high fidelity standard handle a band width to 20KHz or more at full rated power, and slew rate isn't a problem with any of these units. Making it 10 or more times higher than required is of no audible benefit and may increase costs.
On the amp "sound" point, you've left out some requirements given beyond flat frequency response, e.g., inaudibly low noise and distortion, and should note holding other factors equal, particularly matching sound levels to within 0.1dB when running blind tests. The same point always has to be repeated: all an amplifier does is add more voltage, and all more voltage does is make the sound louder. If this would ever change it wouldn't be in audio forum discussions, but might be published in my AES journal and possibly lead to a Nobel Prize in Physics.