The Pentium III is an extremely well-designed chip. The P4 2.0 is also faster than the Athlon Thunderbird 1.4, I believe - although the new Palomino Athlons may be faster than anything Intel has.
When people say "technically superior", what they're basically saying is that the Thunderbird processes more instructions per clock cycle (IPS) than the Pentium 4 does. In order to compete (and to cover up for the fact that the Pentium 4 is poorly designed for current applications), Intel has ramped the P4 up to insanely high clock speeds. This has the pleasant side effect for Intel of selling more chips, because the clock speed is all most people look at.
Hence the misleading new "Performance Rating" system from AMD. This is to cover up for the fact that AMD can't ramp up clock speeds on the Thunderbird processor any higher.
So, it goes like this: At equal clock frequencies, the Pentium III will be faster than the Thunderbird, which will be faster than the Pentium 4.
But it depends on the benchmark, too - the P4 always simply burns rubber at Quake 3's benchmarks, for instance.
Some issues to consider:
The Pentium III Tualatin (I believe the fastest speed is 1266 MHz) is an extremely fast chip, which runs very cool and doesn't consume a lot of power. Whether a chip runs cool or not is important. I'll get to that.
The Pentium 4 is also extremely fast, but it runs hotter and has different power requirements. You will need to buy a new case if you intend to use one. (You also may be able to get away with just buying a new power supply for the case you have, but I don't know about this). The P4 also has the benefit of SSE2. Intel included this new instruction set with the hope that applications might be rewritten for it. No AMD processor has SSE2. To my knowledge, only one application has been rewritten for SSE2, FlaskMPEG (an MPEG-4 video encoder). The P4 is a speed demon on that one.
The Athlon Thunderbird MP currently has the most brute power of any CPU. But this comes at a price: The chip runs very, VERY hot. Don't expect to use normal cooling solutions with one of these. You will more than likely need a case fan, a new HSF (the one included with the processor likely won't be good enough), and possibly even a Card Cooler as well. Intel CPUs have thermal protection - if your CPU ever gets too hot, it will simply lock up until you reboot your computer. Thus, no damage is done to the chip. If an AMD CPU gets too hot, it will simply continue to run until it burns up. Recording is an activity that makes your computer work hard. An Athlon will get very hot doing this.
So you'd need added cooling, causing added noise - not good for recording! And as far as I know, there are many "prosumer" sound cards that the Athlon will not work with.
Look here for a demonstration of how the Athlon copes with heat emergencies. It's very telling.
http://www.tomshardware.com/cpu/01q3/010917/index.html
Lots of people successfully use Athlons for recording, I'm sure - but for that application, I personally wouldn't recommend anything other than a P4 (if you can afford it), or a Tualatin core Celeron or Pentium III.