Of course, you cannot discuss bit rate without mentioning something about sample rate. When you record to analog tape, you are recording a continuous representation of a waveform. That's the nature of analog. However, it is not the nature of PCM digital audio. At a sample rate of 44.1k, you are taking 44,100 individual snapshots of the waveform every second (44,100 samples per second.) This is why higher sample rates are better, because you get a better representation of the waveform. Your sample rate also limits the range of frequencies you can reproduce, but that is another discussion entirely.
Now, when you talk about bit rate, you are talking about the size of each sample. Digital audio is encoded in binary code, which is a series of one's and zero's. So, the bit rate determines how many ones or zeros each sample will have. At 16 bits, each sample is represented by 16 ones or zeros. At 24 bits, then, you have 8 more ones or zeros per sample. As you can probably imagine, the amount of combinations of ones and zeros is greatly increased at 24 bits. Thus, you get a more accurate representation of the sample. You have increased resolution, which ultimately means greater fidelity.
This increased resolution is often more noticeable when it comes to changes in dynamics. For instance, at 16 bits, a fadeout might at some point seem to just fall off a cliff! In other words, you might hear a nice, smooth fade for a while, but then suddenly the signal is gone. At 24 bits, however, the fade might sound infinitely better. In fact, many engineers feel there is a drastic difference in audio quality between 24 bits and 16 bits. You may have correctly inferred from what I said earlier that higher sample rates mean larger frequency range. Similarly, higher bit rates mean larger dynamic range. Therefore, musical styles with greater dynamic range, such as jazz and classical, benefit from higher bit rates. With Rock music, on the other hand, which is often heavily compressed anyway, it may not matter as much.
Now, back to dither. Even though dither does add a small amount of noise to the signal, there is a second algorithm called noise shaping, in which the dither noise is re-EQ'd out of the audible range. However, it is more important to understand the true purpose of dither. When you reduce the bit rate from 24 to 16, you are reducing the resolution of each sample by 1/3, basically truncating the last 8 bits off of each sample. This can reduce the dynamic range, and generally degrade the sound quality. When you add dither, it does something akin to rounding off the last few bits to make up for the 8 bits you are losing. In other words, it is designed to allow you to preserve at least some of the dynamic range and overall sound quality when you reduce the bit rate.
The best way to A/B between a dithered and undithered mix would be to listen to a fade-out. The dithered version should sound smoother.
Of course, these differences are, admittedly, rather subtle, and I do know a few engineers who think it's all hogwash. The ideal solution to this problem is a 24-bit digital audio standard, hopefully at a higher sample rate as well. I believe this will happen very soon.
A few years ago, I had a friend who would do all of his sessions at 16 bits, reasoning that it would keep him from having to worry about bit-rate reduction and dither. I actually bought into this for a while, and was doing the same thing. Then one day I read that Pro Tools does all of it's processing in 24 bits. This means that whenever I was using a plug-in or doing a bounce, even a gain change with a fader, it would increase the bit rate to 24, then go back down to 16. So, obviously, there was absolutely no advantage (and, actually, a big disadvantage) to doing things that way.
Then, as I studied digital audio even more I found out that every audio calculation you do, even in the 24 bit range, briefly adds bits to the end of each sample, then cuts them back off to return to your session bit rate. (I had taken recording classes back in the analog days, so I never learned this until I started teaching it.) That's why, for instance, you can set Pro Tools to use dither for audiosuite plug-ins. The consensus among audio engineers, thankfully, is that, even though there is a big difference in sound quality between 16 and 24 bits, once you get above 24 bits, the differences are much less severe, so these recalculations do not adversely affect the sound quality.
One other thing to remember about dither: Since it does add some noise to the signal, you usually don't want to do it more than once. That's why it should be the very last thing you do while you are performing a bit-rate reduction (such as a bounce to disk operation.)
When mixing an album, it’s best to stay in the 24-bit realm, even when you bounce down to stereo mixes. Then, you should take your 24 bit stereo bounces back into your DAW or mastering program, and apply EQ, multiband compression, limiting, etc. in order to get consistent levels and EQ between each song. Finally, insert dither on the master fader and bounce each song to a 16-bit file. In other words, if you are working on an album or EP project that will be mixed and mastered, you should stay at 24 bits all the way through to the end of the mastering phase.