The most analogous (get it!) comparison to the analog world would be dynamic range.

Each bit is a binary number, so with 16 bit audio you have 16 levels of binary numbers, I'm not going to do the math right now but that's like 65,000-something binary levels.

At 24 bit you expand that to have 24 levels of binary numbers, which becomes something like 16,000,000-something binary levels.

So you're looking at greatly increasing the accuracy of the sampled data.

With a nice setup and 24-bit soundcard you should be able to hear a difference.