First of all, there is (edit: was) a potential terminology issue going on here. Please understand the following: Dithering noise is not something that exists, but something you can add to get rid of something that exists. That something is often called the quantization noise caused by rounding errors during the sampling or the conversion back from 24 bits to 16 bits. It sounds like an aggressive noise that is most heard in high frequency range.
A typical way to display a rounding error in a signal, is to think of a picture. Pictures also have bit depths in colors. Imagine you took a picture with your digital camera which has a nice color gradient from blue to green in it. To actually see this gradients, you need millions of colors between blue and green. Now imagine you change the bit depth from e.g. 24 bits colors, to 16 colors. (which is 4 bit.) by truncating the 20 least significant bits. You will suddenly see that instead of the gradient, all millions of colors are reduced to the nearest of the 16 colors in the 4 bits pallette, in this case just blue and green. While this idea works fine on images with sharp edges and little colors, it doesn't work on gradients because it introduces a sharp lines between them at the place the color changes from blue to green.
If you add dithering, there is some noise added to the signal. (Or, in this case, the picture.) Therefore the color values are changing slightly from their original value, each pixel. This means that at the point the sharp color-changing line was drawn, there is now a pattern of the two colors blended together, since the noise introduces a few green pixels on the blue side (because the random dithering made them just a bit greener), and a few blue pixels on the green side (idem) and it looks - well, still terrible, but the sharp color transition is gone.
Exactly this happens in audio as well, Try recording a 16 bits signal, (make sure it is loud enough, so get it to 0 dB FS and add a compressor if necessary) and truncate the least significant 15 bits, reducing it back to 1 bit. You get a 1 bit signal that you really can't use. There is no way of recognizing the original back. Now (and this is really miraculous!) mix white noise in the least significant 15 bits (equivalent to -6 dB FS) and then truncate them to the one bit signal. You will hear the exact original signal, mixed in a huge LOT of white noise. But you can actually hear more information from the original than without dithering.
Think of it this way: If you dither in those 15 bits (i.e. add a random value), you actually add information to the 16th (most significant) bit. See it this way.
- If you truncate those bits, you round all values to 1 or 0 by checking whether they're above or below the sharp line of 0.5.
- By adding dithering noise, it could possibly be that a 0.4 turns into a 1 as well, and a 0.6 turns into a 0.
- The chance of a 0.4 becoming a 1 is bigger than the chance of a 0.1 becoming a 1, since you add a random number.
So instead of drawing a fixed line saying all values below 0.5 will be 0, the rest will be 1, you state: All values will be 1 or 0. 0.9999 has most chance of turning into a 1; 0.0001 has most chance of turning into a 0. That is what dithering is.
** The additional question **
You asked: what is the difference between a 16 bits recorded signal, and a 24 bits recorded signal with the last 8 bits truncated? The real (theoretical) answer: there is no difference. If you don't plan to dither, and you are not increasing the signal volume to do something useful with the 8 least significant bits, you can as well record in 16 bits. BUT: There will be a big advantage if you dither.
This said, the real-time rounding mechanism of 16 bits recording equipment is often a bit different from rounding by truncation, since it deals with analog data, so there is still some kind of random bit-flipping in the converter going on, which acts a bit like dithering. Whether this is something you want in your signal depends on your equipment, your ears and most important, your opinion. :) I'm not going into detail about this though, but the conclusion is that you get more quantization noise from truncating a 24 bits signal than you get from sampling a signal to 16 bits. This is not going to be my point anyway.
This sounds disappointing, but the reality is, you are asking the wrong question. The real question should be: What can I get in my 16 bits master when I record in 24 bits, what I can't get if I record directly to 16 bits?
So before you start asking the other obvious question: Yes! Still, do dither! It is the only way to move as much information as possible from the 8 least significant bits of the 24 bits signal, into the 16 bits signal. Think of the photograph with 16 colors. Nobody would recognize it if it wasn't dithered while the bit depth was still high. Also, your camera can't dither by itself, so it can't take recognizable photographs with 16 colors. To get the best: record (i.e. take picture) with high bit depth, dither, reduce. :)
Now: the real bottom line.
Recording in 24 bits gives more detail in low dynamics. Details that you can add to you 16 bits master by dithering, that you can't get when recording in 16 bits directly.
Don't ask yourself the question: Should I dither?
Instead, ask yourself the question: Do I want to record in 24 bits? And always dither if you do, to actually get the advantage of it. :)