The easiest way, like joojaa suggested, is to duplicate and expand (i.e. convert the stroke to a path). Then you group the two so you can transform them together.
If the path is closed you can play with the 'align stroke' parameter (stroke panel) to have it be internal or external.
It's because what you are seeing is merely a low resolution preview of the brush.
The actual brush isn't applied to the path until you lift the stylus. There's no way to change this behaviour in Illustrator's brush tool. It will always apply some level of smoothing after you release the stylus, even with the fidelity set all the way to accurate.
Perhaps if ...
The settings are counter-intuitive.
You want the Fidelity setting to be low, not high.
The devs are engineers.. in math a high fidelity means less accurate.. to common, non-engineers or non-physicists.. a high fidelity means more accurate...
I, personally, set the Fidelity to about 0.5-1px and the Smoothness to 1-2%.
I would also point out that brushes are ...