Couple of things come to mind.
When you buy an amp w twice the rated power, the power supply in that amp is generally much bigger than in the smaller one.
As was mentioned in a post above, a bigger amp playing at the same volume as a smaller amp will sound more dynamic - provided they are built to the same quality level. I tend to believe this difference is connected to the size of the power supply, more so than the power (watt) capability of the amp. The Hiraga Super 30 watt class A amp I built, has 192K uF in the PS, with a 500VA transformer. People build these amps with insane amounts of capacitance, but for this built, I went with this. Next time I build this amp, I’ll be building it as a dual mono design with greater capacitance in the storage so I can hear first hand how that translates into real world listening experience.
I recently built a single ended EL84 based amp. It’s about 3 watts. Love the sound, but want a bit more usable volume. So I’m about to built a single ended EL34 based amp that will get me approximately 6 watts. Double the power. Different tubes and circuit design, I know, but I’m just curious how that 6 watts will present compared to 3 watts. 3 watts don’t mean a hill of beans if your amp is pushing 200watts into the speakers, but going from 3 to 6 is doubling the power. Thankfully, going from 3 to 6 watts is a heck of a lot less expensive than going from 150 to 300 watts :)