The ability to drive difficult loads. How robust is the power supply? etc. You can make an amp that sounds acceptable at low levels driving easy loads. But, when you really need power (not volume), it is like trying to get large flows of water through a small diameter pipe. not going to happen. That is why power supply design to date is still extremely important.The ability to drive difficult loads doesn't seem to be helpful to any amplifier, because its easy to see on the spec sheet and by listening to the amp on a hard load that its making more distortion. The problem is, such distortion is audible as brightness, harshness and a loss of detail. You can take any amplifier and just by putting it on a higher impedance/easier-to-drive load, it will immediately sound smoother with more detail. That is true whether its tube, transistor or class D. You can both hear it and see it in the specs- this is one of the few areas where the ear and the specs agree.
I do agree though that robust power supply design is one of the things that set high end amps apart from the mid fi.