Although watts actually are a pretty decent way to predict the loudness of an amp, it's worth understanding what the listed wattage of an amp means. Remember that an amp is designed first and foremost to amplify the input signal, and that for many applications, it's desirable that the amp do so while keeping the signal as clean as possible. So an amp's wattage rating is traditionally listed as the amount of power it can produce while keeping the total harmonic distortion (THD) of the output under a certain percentage.
But maybe the guitarist doesn't care about keeping the signal pristine---in fact, maybe he'd like the amp to produce some distortion, maybe even a lot of distortion. In that case, he's going to be running the amp at higher wattages than its rating and thus it will seem louder than its rating would imply. It matters, too, whether the amp uses tubes or solid state transistors, because tube distortion is often thought of as being more pleasant than solid state distortion. This means that a guitarist playing through a 100 watt tube amp may turn it up to get distortion from the amp, while a guitarist playing through a 100 watt solid state amp may rely on pedals for his distortion and leave the amp turned down to avoid the unpleasant solid-state distortion.
You mention the Beatles playing through 15 watt Vox AC15's, which are tube amps. In concert, they turned those amps way up and got a fair amount of distortion from them, meaning their amps were producing way more than 15 watts of power. Perhaps the 15 watt amp you mentioned as being underpowered was a solid-state amp?
As far as amplifying other instruments is concerned, each instrument has its own specific needs, and so its hard to compare wattages across different instrument categories. Keyboard amps, for example, are designed to be as clean as possible, so a keyboardist is not likely to turn an amp up beyond its listed rating. Bassists, as a general rule, require higher-rated amps because their lower frequencies require more power to produce a similar volume. For example, in a typical rock band, the guitarist might play through an amp rated anywhere between 50 and 150 watts (if you're planning on cranking your amp up to AC/DC-type distortion levels, btw, 50 W is plenty loud), while many bass heads are rated in the 300-400 W range. In my own case, my guitar amp is a Fender Twin Reverb rated at 85 watts and my bass rig is powered by a Crown K2 power amp which produces 1,600 watts in bridge-mono mode. I can tell you that the Twin absolutely keeps up with the bass rig in terms of volume---but then, I want to distort the guitar amp, and I want to keep the bass amp as clean as possible.
In any event, wattage is not bad as an indicator of an amp's loudness, so long as you understand what it does and doesn't actually mean.