Does the power of an amplifier really matter?

Excuse my ignorance, but I am trying to understand the meaning of amplifier power. A doubling of amplifier output power results in a 3dB increase in sound-pressure level. 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)? I am not sure that is the case. Is it that a higher powered amp delivers a given amount of power more cleanly? Not sure if that is true as well. Can anyone educate me on this fundamental property of amplifiers?
It depends greatly upon the design of the amplifier and the characteristics of the speakers that are used with it.

In many cases, it is only the first watt or two which make the most impact on the sound. However, as playing gets louder,particularly with some inefficient speakers(and some with difficult impedances) the amplifier can become strained, and can eventually go beyond its ability to drive the speaker cleanly.

Current delivery ability of the amp can become increasingly important in these situations, and also when deep bass response in in the equation.

Since it requires about double the amp power applied, to increase most speakers SPL by 3db/1 meter, it is apparent that higher amp power can be useful to extend the SPL of many speakers up to their maximum SPL capability.

Additionally, dynamic headroom may play a factor in this, and many amp makers will include a dynamic headroom spec, which means that their power supplies are robust enough to provide "bursts" of power beyond their rated power levels, to handle difficult musical passages.

The more modest your listening SPL demands are, generally the less demanding you are of amp power, unless your speakers present a particularly difficult load for the amplifier to handle.

I realize that many other technical factors are present, but I'm trying to explain it in simple layman's terms.
"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!
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.
Number1, yours is a dilemma common to many of us.

There are no easy answers in spite of what the measurists would have you believe. The ability to maintain voltage in the face of low impedances...IOW the ability to deliver high current...probably is one factor, and the amp's sound at and beyond clipping probably is another.

Amplifier power is just a beginning, and probably not a very good one.
I just wish I could get a handle on that nebulous amp attribute that allows a 100W amp to shine over a 200W amp
Or, vice-versa of course. Very robust power supply /rectification, matched output devices, carefully chosen components, application of feedback, etc, etc. So many things that are NOT visible to the naked eye...

BTW, regarding spl's, in the near field, ribbons lose about 3db per doubling of distance vs 6db for cones. A difference of 20db spl b/ween sound 1 and sound 2 means 2 is ten times louder than one. To achieve an extra 20 db difference you need an extra 100W of electrical energy. Accordingly, 250W should achieve ~24db. If you want 40db headroom (ie 100 times louder) you need 10.000W. Classical music often has 30db differences. Under normal conditions, that would require 1000W over the reference low passage.
Depressing isn't it:)
Fortunately, we usually listen at low spls and consume milliwatts of our amp's power, so there is headroom enough even with a 50-100W amp -- or less -- for the forte.
So when the cannon goes off in the 1812 Overture, you guys don't use the subwoofer to take over? Does any body know what's the frequency of that canon in the Overture? Your full size speakers can deliver such power and such low frequency? I am just in awe, and I am afraid to find out what speakers do you guys have b/c I may inherit a bug that I cannot afford. However, please do tell, and I am all ears. :> )
Some "low powered "amps will power speakers that simply don't work with other amp designs with much more power. The vintage Levinson ML9 (get that right?) was a pure class A monaural design with a whopping 25 watts into 8 ohms....and 50 into 4.....and 100 into 2....and 200 into 1...and it WAS rated at 400 into 0.5 ohms....

I would pick up a pair of these if they show up in good condition at the right price in a heartbeat.
The levinson ML-9 is Class A/B and it was 100 watts per channel in 8 ohms. It is the same amp as the ML-3 which had 200 watts per channel. You are confused with the Levinson ML-2 which were mono 25 watts class A and were used with Quad speakers mostly at the time.