Sonic quality. 24 bit has much greater sonic quality. The difference is startling to me. I wish it were not, because when I used to record 16 bit I had that many less steps. But once I tried 24 bit, there was no way I could go back. This is not to say I cannot get a great 16 bit recording, but all things being equal, the 24 bit recording his more color and tone. It sounds better.
I'm afraid you can't put that (more colour and tone) down to the number of bits used. There must be some other factor involved in the comparison.
A 16 bit recording and a 24 bit recording are the same until you start dealing with levels of less than -96dB (off the top of my head). Then the 16 bit recording runs out of bits but the 24 bit recording carries on down to -144dB (again, off the top of my head). Don't make the mistake of thinking that 24 bits gives you more measurement of the audio across the same extent as 16 bits and it's therefore 'more accurate because the intervals are smaller'. Imagine you have a 16 bit ruler and a 24 bit ruler - the 24 bit ruler is longer, with the notional graduations the same space apart as the 16 bit ruler (though the graduations have a log base just to make things more complicated).
In practice this means that if your preamp has a range between noise and overload greater than 96dB, then you will need 24 bits fully to capture that range. However, bear in mind that at the bottom of that range, you'd better be listening in a very quiet environment to hear it - even with most headphones. More usefully, you can record in 24 bits so that the highest peaks going into the A to D converter never go over about (say) -12dB (or much less in fact), and you'll still be able to convert it back to analog without losing the low level information which will be stored in the part of the bit range than 16 bits doesn't have. That still doesn't absolve you from considering the signal to noise ration of the analog stages of the system - if, in order to record at those lower levels, you are under-running your preamp and its design is such that it's not working at its optimum signal to noise level, you could end up with a worse result.
As always therefore you have to have a broad grasp of the theory - then throw away the rule book and listen with your ears. (Heh, but don't let your ears tell you something that the rule book says is impossible!)