My 2 cents
First off, welcome to the nasty world of digital audio! And secondly, welcome to the even nastier world of marketing-speak! Okay, that's unfair. They both offer (usually) audible increases in quality (which the manufacturers are quick to point out), but they also bring with them difficulties (which the manufactureres hope you won't hear about)!
I guess the quick Linus answer is, "It's what you make it, Charlie Brown." I mean, let's face it, in the end, we're stuck with 16bit/44.1kHz, and since CD's sound pretty darned good, it can't be THAT bad, can it? Let's try it this way: if you took two 2-track DAT machines, one at 16bit/44.1k, the other at 24bit/96k, then dithered the 24bit jobbie back down to 16bit/44k, it should sound ABOUT the same. The 'about' has to do with things like dither algorithms and such, but they should be pretty close, overall. So in this particular application, it would make no difference at all.
The problem is, we don't have theoretical converters, we have real ones, with noise, nonlinearity, and gain/offset errors. Let's break the two parts down, sample rate (44.1kHz vs. 96kHz), and resolution (16bit vs. 24bit).
Sample rate is pretty easy. The faster you sample (assuming your error rate doesn't go up), the higher a frequency you can capture. A natural consequence of this is that you can also capture faster transients (snare drum attack, pick strike, vocal noises) better. This helps you twice. First, you get more information to work with when you record, so you have more information to work with when mixing. Second, the slopes of all your digital filters can be more gentle (natural) and thus subsequent processing won't make the high end sound as forced. The two downsides here are bandwidth and sample-rate-conversion. Bandwidth simply means you need more than twice the hard-disk space, transfer speed, and processing power to handle the same audio stream for 96kHz vs. 44.1kHz.
Sample rate conversion is something a lot of people don't like to talk about. An exception here has been Sonusman, who in the past has spoken out vehemently against all sample rate conversion. The problem (especially when going from 96kHz to 44.1kHz) is that there are 2.176 samples in the 96kHz track for every one sample in the 44.1kHz track. Even if you used an 88.2kHz sample rate, just throwing out every other sample will not necessarily yield the most accurate 44.1kHz output. Throw in the non-integer multiple, and you need to start using math to "guess" what the sample was that was in between (the .176 th of a sample). That means that you are usually off by some amount, and someone with golden ears somewhere is going to say they hear it. Will you? twist the knobs, Charlie Brown! You'll need to see for yourself, as only you can tell if you can hear the difference. I can't afford the bandwidth, so it's a non-issue for me. Either way, if you can afford the higher sample rate, you're usually better off to use it.
Resolution is a very different beast. First off, esactun was right about both topics. So with higher resolution, the finer a detail you can resolve between different input amplitudes. That's how it applies to recording, but it's even more vital to the processing that happens during mixdown. We all know that if you multiply 4.2 * 3.2, you get 13.44 (at least my calculator knows that, anyway), but if I limit myself to that one decimal place of 'resolution,' I can only say I have 13.4 . Now what if I really want to know 4.2 * 3.2 * 2.2 * 1.2 ? My calculator says the answer is : 35.4816. But lets say I do it as three separate multiplications, and only keep 1 decimal place of resolution? We've already got 4.2*3.2=13.4, so now it's 13.4*2.2*1.2.
13.4*2.2 = 29.48, so I call it 29.5, now I've got 29.5*1.2 = 35.4. But if I round my original answer, 35.4816, to 1 decimal place, I get 35.6! So I'm off by (in effect) a whole bit! And that's only for 3 operations! What if I need 8 operations on a single sample to calculate the EQ in
my Waves Q10? I could be off by even more! So they usually keep much higher precision inside the plugin (so you'd actually GET 35.4816, rounded 35.6), and only do the roundoff at the end.
But what if you use 10 different plugins on a single track? Bingo! Same problem, only this time you don't have the extra internal precision to save you. Here's where that high resolution REALLY helps you out! Since we're only keeping 16bits of info anyway, we can afford some error down in that 24 th, 23 rd, or even 22 nd bit without hurting anything very much.
So what's the cost? Well, about 50% extra bandwidth, just like the increased resolution. But there's another problem. All converters are NOT created equal! I believe Bob Katz has said that he has yet to hear a 24bit converter that yields as much useful information as his own (super-top-of-the-line) 20 bit converters. And others have said that some poorly designed 24bit converters actually show residual noise (not from the input signal) up around 16bits!!!
I've spent very little time researching 24bit converters, since I'm happy with the 20-bit units in
my Yamaha 01V. Are they the best? NO WAY! Do they work for my crappy home recording? YEPers, baby! Can I afford even 2 channels of Apogee or Lucid conversion right now? Nope, but it's not that big a deal to me right now. At some point you've just gotta stop buying gear and start recording/making music. That's where I am right now.
Okay, last thing here is DITHER, that nastiest of nasties!!! It's just like sample-rate conversion, only now you're trying to figure out what series of 16-bit samples will sound the most like the series of 24-bit samples you started with. There are a lot of good discussions of this here, and my own opinions are just that (I can't really hear the difference between the dither algorithms I've heard so far), so I'd just suggest hitting the SEARCH button for more info on dither.
I know this has turned into a MAJOR sermon at this point, but I hope I've cleared up some of the issues surrounding digital audio and its 'no-free-lunch' behavior. If you have specific questions about anything I've said, please ask away. And if this got WAY too technical (I never do that

) I apologize, and I think I can put any of this in to as layman's a term as you'd like (I hope

)