Generally, if you use cheap gear and/or use a lot of synths/sound modules, you'd probably never notice the difference other than having less tracks in a project before you run into problems.
If you use quality gear, and record lots of "real" instruments/vocals, expecially strings/cymbals, you will probably notice a difference in clarity/sheen/etc, or if you have a lot of tracks in a project the mix will probably sound clearer.
The second biggest single change for better sound would be to use 24 bit options instead of 16, (more headroom with less loss of quality) and if using a DAW that allows it, enabling the mix for 32 bit floating point. This last can be done even with 16 bit hardware, and keeps mixing artifacts to a minimum.
The BIGGEST improvement is to learn each and every piece of gear you have, and how to get the best possible performance out of it. If you do this all the way through your signal chain, you won't believe the difference... Steve