16 bit has a deeper color gamut...more colors. Many people won't really notice a difference between the two...but more colors is better...especially when you are doing a lot of editing.
The more colors you have...the smoother the transition from one color/tone to another.
When you edit an image and change the colors or tones...some of the pixels have to be changed...and that means that some of the intermediate pixels are lost (because you are not adding more).
I read an article a while back, that explained it much better than I am doing. Think about the histogram of an image (in photoshop). Open up the levels dialog and make some changes...and hit OK. Then open it up again. It will probably have gone from a smooth graph to having some lines missing. Those are the missing pixels that have been changed/lost. It may not show in the image (but it might) but those 'gaps' will end up as lines where it should be a smooth transition.
If you have more colors...then the gaps in the graph will be smaller...which means that your transitions will be smoother.