How Much Difference Does a More Powerful Amp Make?

For inefficient speakers with low impedance drops a more powerful amp can make a dramatic difference. For your average speaker you may perceive better or tighter bass and more "ease" at louder volumes. But then again you may perceive nothing different at normal listening levels and sometimes lower powered amps do indeed have better sound than their big brothers.

Amp rated at lower continuous power might have higher momentary power. Average power needed for music is only few percent of peak power unless you listen to sinewaves.

Power specifications give only general guidance but all depends on particular amp design. Chances are, when comparing amps from the same line/company, that higher rated amp will be louder but 15.7% difference in loudness might not be worth extra money.

Comparing to a car is a little misleading since 10x power is only 2x louder (exponential scale).

The point has already been made but here it is in a cute form.
If the first watt sounds bad why would you want several hundred more of them?
Quality is the key and the usual recognition that some speakers really do need high current amplification to sound best. Essentially I am just agreeing with most of the comments thus far.

The W7 is the sweeter sounding of the two amps . I would never trade finesse for more power , and although the W8 may sound more powerful we are only talking about a 2dB difference .

A couple of years ago Hi-Fi+ published and interesting review, where they compared three Conrad Johnson's power amps- LP70, LP140, LP280, which use essentially the same design, circuit and type of output tubes, the only difference being, as you could probably guess, an output power-70, 140, and 280 Wt. respectively.
Without going into much detail, the bottom line was, that in the same system (actually with some fairly efficient speakers), every more powerful model sounded better, than the one with the lower output. LP 280 was an overall winner.
I completely concur with the above based on my personal experience.
Power of the amplifier has very little to do with the loudness, or efficiency for that matter. It has more to do with the ability of the power amp "to control" the speaker, allowing it to play at it's full potential. Obviously, it has direct impact on how the dynamic aspects of music are being handled.
In essense, you can make 10 Wt. amp to play almost any speaker loud enough, but it won't sound good doing it.