The first issue is 24-bit use more storage capacity than the 16-bit system, it means that the quality of the recording is better in the 24-bit system because more numbers are used to represent the original analog sound. As you know the recording process in Digital Audio world begin with the conversion from analog signal (the sound from mic or your instrument musical) to digital number (0 and 1), because computer only recognize and process numbers (sample). This process is called A/D Conversion (Analog to Digital).
When you have 200 and 400 pieces of small playing blocks or lego for examples and you are asked to form a dinosaur from this blocks, which one will be more looks like a dinosaur, ofcourse the latest will be. 400 Block will give you better result because it will give you more chance to add details in the dinosaur shape, for example if you only use 200 blocks maybe you have run out of blocks when you want to add a little detail in the nose of the dinosaur. The above example is only a metaphor to give you simplified version of the theory. So in this first issue 24-bit system will let you capture more dynamics in your recording than 16-bit system. I use Sound Forge with M Audio Sound Card for capturing vocal in my recording studio and with 24-bit recording session, it sound very crispy.
The second issue is every process in the Digital Audio Workstation world result in more numbers, for instance equalization, compression or the reverb delay (predelay) of a digital audio signal will result in more numbers used. As an mixing engineer we shoud notice this. The calculation is determined by the audio plugins you used for example FL Studio / Fruity Loop plugins, computed by the your computer’s CPU (or a DSP chip) and the output is stored in the hard disk.
We will use a simple math to show this. Divide 96 by 23 the result is 4.1739130434782608695652173913043 (32 digits numbers), now you are asked to fill the result of the calculation to this 16 column of table, you will omit the rest 16 digit and only can use the first 16 digit. The case will be different if you are given 24 column to fill, you only need to ommit 8 digit, and maybe these 8 digits is not as important as the previous 8 digits. In the recording world the first 8 digits (after the 16 digits), for example can refer to your reverb processing data. So if you omit this it may result in muffled reverb sound. So in this second issue 24-bit will give you more transparent sound than 16-bit system.
The conclusion is whenever possible, use 24-bit system, it will have better result but use more storage. But please note that the quality of A/D converter itself will also affect the quality of the result. A professional 16-bit A/D converter (for example from Digidesign or a Roland VS system) will beat average consumer 24-bit A/D converter sound card (may be only suitable for gaming or other home activities).