In these days of oversampling converters, bit depth wins over sampling rate (IMNSHO, anyway). These days, the analog artifacting of old due to the antialiasing filters is pretty much a thing of the past.
Recording at 24 bits allows for more flexibility in setting levels, and some folks would describe this as "more headroom". Truth be told, in the digital world, nothing provides more headroom: you can *always* turn the knob up until it crunches! However, 24 bit encoding pushes the noise floor down somewhat, so you can record at more conservative levels and still get good signal-to-noise (in this case, signal to noise+distortion, because you're primarily concerned about the distortion characteristics of the A/D-D/A chain for low-level signals).
16 bits gives reasonable dynamic range, and preserves reasonable noise characteristics for low-level signals, as long as you use it all. 24 bit encoding provides a much more benign environment for low-level signals. Just listen to your reverb tails in the two environments, as the reverb decays: that's my classical test for it with good converters. Even I can hear a certain "graininess" in the 16-bit stuff that is less obvious in the 24 bit version, and I'm dadgum near deaf these days.
To get best linearity and best S/N at 16 bits, conventional wisdom holds that you must record right up at 0dB on the peaks, with no margin for error. With 24 bit, you get equivalent or better performance, even if you back off to -6dB or more for the peaks. It makes for a much more relaxed tracking environment, when you don't feel that you have to max the bejeezus out of everything- much more of an oldtimey analog feel, when the tape could "handle it"...
If that's more headroom, it's a psychological thing, *not* a circuit thing as it was for analog tape. I don't care: I'll take it, and I'd rather be more relaxed any day!
However, as in all things, there's a caveat: some "24-bit" converters out there are really only good for 15-16 bits of actual resolution and linearity. We can thank the markerters for that one... In which case, your "extra headroom" is illusory, because those low bits are just junk.... Check the specs: if the S/N and/or dynamic range of your 24-bit converters is 90dB or less (and there are some that are much worse, and are still marketed as 24-bit) you might as well just record at 16 bits: 24 ain't-a-gonna help, and you're going to have to fish your signal up out of the noise floor anyway, and work as hard as you would with simple honest 16-bit conversion to get equivalent results.
Confusing? You bet. However, consider this: there are some 20-bit converters that actually produce more usable linearity, which controls both S/N+D and dynamic range, than many of the low-cost 24-bit guys.
Here's an attempt to clear up the confusion I just created. My example of the AK converters above stands. They are "24-bit" converters, but produce essentially 20 bits of usable linear range. I'm happy: I get 4 bits more resolution than a "perfect" 16-bit converter, so I can back off 2 bits worth (6dB) and still have 2 bits better resolution and linearity than the ideal 16-bit example. And less ulcers worrying about crunches and overs! Make sense?
Somebody with cheapo pseudo-24-bit converters that really are only good for 16 bits, and who backs off 2 bits worth, will be recording only 14 bits above their noise floor. And that's a long way from optimal: that's a _lose_.
So what are your converters good for, in terms of _linear_ range? That's what makes the "extra headroom" magically appear. Those 8 extra bits don't help much unless what they contain has some relevance to the _signal_, no? Caveat emptor... Hope that helps.