The end user will almost always be seeing 8-bit, and your video sources are
probably 8-bit as well, unless you have a very nice camera. 10-bit video
offers more detailed shades of color. For example, if there is a very dark
area in your image, 8-bit might just show black, whereas 10-bit could show
some detail. Of course, if you deliver your final result in 8-bit, then you'll
need to somehow brighten those areas to bring them into the 8-bit range.
It's about the amount of data you have available to represent your image.
Let me try a metaphor. Bit depth is like image size. So, for example, let's
say you shoot a video at 1080 and you are delivering it at 720. It makes
sense to stay in 1080 until the very end, because all your details will be clearer,
and the approximations more accurate as you process. Then, at the final
output stage you can make the compromises inherent in reducing the size.
If you have 10-bit footage (i.e. DigiBeta), and/or you're planning to do a lot of color correction,
effects, and gradients go with 10-bit. You'll get better results.
On the other hand, if you're a beginner working with basic 8-bit footage,
stick to 8-bit. It will work just fine for now.