The improvements can be subtle, especially if you're coming from an excellent receiver to another excellent separate amp. You would need to really know the material you are playing very well to distinguish.
In my case, from a flagship 97 Onkyo to an Acurus separates, the difference in power rating is already telling. The Acurus has twice the power - a good 3db improvement for the same volume setting. And anything louder is always perceived to sound better. The stereo image is more spaceous and precise courtesy of better channel separation. The instrumental details became more apparent, courtesy of a higher signal/noise ratio and lower THD/IM. The bass is more defined, thanks to a much higher damping factor, from 50 at 60hz to 500.
I guess the separates you are contemplating at really has to have superior specs to begin with. I will not discount the possibility that after reading the superior specs, you may have a psychological "power of suggestion" thing to expect better sonics. But psychological or real, the important thing is you perceive a difference. This hobby is almost always about perceptions.