Does the power of an amplifier really matter?

"So, if I listen to my stereo at a volume of say, 70 dB, what is the difference in a 100W amp vs. 250 W amp? If I listen at 70 dB, is more power delivered to the speakers with a 250W amp vs. 100W amp (resulting in better driver control)?"

No. A speaker needs a certain amount of power to play a particular note or tone at 70 dB (in your room, at your listening location). That's exactly what the amp is delivering at that moment, whether it's a 100w or a 250w amp.

The advantage of a 250w amp is that music, unlike a test tone, isn't at constant loudness. Occasional peaks can really drive up a speaker's need for power. With a more powerful amp, you're less likely to hit its limits or start to hear noticeable levels of distortion when the cannon goes off in the 1812 Overture, say.

"Is it that a higher powered amp delivers a given amount of power more cleanly?"

If you're talking about a large amount of power, that may be the case. Most of the time, however, your speakers aren't demanding very much of your amplifier, and even a small amp will be able to deliver a clean signal.

Keep in mind that an amp's power rating is only a very crude measure of how powerful it is under the range of conditions it would face in a typical music system. Someone who knows what they're doing can do much more meaningful measurements, but the measurements you see on the typical spec sheet aren't particularly meaningful. The best way to tell whether an amp is powerful enough is to bring it home and try it out in your system.

Several personal points here:
1-Yes, every 3db requires a doubling of power. However, it takes +10db to double the perceived volume.
2-70db is a tad too quiet for most people. Most speakers are putting out 1 watt to produce around 89db. Simply to double the perceived volume would require about 10 watts (99db).
3-Finally, keep in mind that although it doesn't seem like a big deal to exceed an amps max rating just for a short period of time, it can be. Some very nice amps handle dynamic headroom really well, while some lessor quality units start distorting rather quickly and don't recover for a while. To put that into perspective, compare a 7 watt cary amp to a 100 watt klh receiver. Both playing at 5 watts will sound drastically different. In light of what Pabelson stated above, the 1812 overture may not be as loud on the Cary, but it sure will sound better and handle the peaks with no noticable problems.

As the owner/operator of Magnepanar speakers, I have learned a thing or two about amp power. I have used various amps from 100 to 600 watts/channel (4 ohms) and for every step of increased power the sound quality has improved. This has little to do with how loud the speakers play: the 60 watt amp makes plenty of volume. I don't think that it is the power per se, but rather some other characteristic of an amp which just goes along with the ability to deliver a lot of power.

To Pabelson--WOW, a highly intelligent and thoughtful response. I was thinking along the same lines but you said it first and VERY well.

To Elevick--if your '...around 89db' is intended to address a sort-of-average speaker's sensitivity, don't forget that those sensitivity measurements are done at 1 meter. At 2 meters, that SPL is down 6dB (I BELIEVE; someone pls correct me if that's wrong)...which, of course, reinforces your point about power requirements.

Overall, IMO the 2 most-important determinants of power requirements are speaker sensitivity and desired listening levels. There is maybe 10 deciBels difference in typical sensitivities in popular speakers*, and there's probably more like 20dB difference in typical average listening levels. Combined, that's a 30dB difference which is HUGE.

Eldartford is correct IMO--'sufficient' power gives you a sense of ease, of effortlessness, in playing dynamic music.

* I'm NOT including the rare examples of those speakers in the high 90s and 100s. If I did, that difference might be 20 or more deciBels!
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All of these replies are excellent, and I thank you all for sharing your knowledge. It appears to me that--yes, more power can give you more dynamic headroom; a sense of ease during a John Coltrane solo hitting the highest octaves at high volumes--or Paul chambers hitting the lowest octaves in terms of articulation of tight bass--but, there are probably more important attributes of the amplifier that relate to delivery of that power, and a clean delivery results in impoved micro- and macro-dynamics. So a 100W amp can be much more articulate, dynamic, and have better imaging ability than a 200W amp. The point is that one should not make a decision based on amp power but on musical presentation--which applies to all audio equipment. I just wish I could get a handle on that nebulous amp attribute that allows a 100W amp to shine over a 200W amp, unless that ability relates to a clean power output alone.