The phosphor-bronze (looks like brass) brush is for scrubbing the bore down, removing residues in the grooves of the barrel. On a commerical cleaning rod, you will note that the shaft (and brush) can rotate freely....this is so that the bristles can get into and follow the grooves thru the rotations of the riflings. Bore cleaning solvents are used with this, and lead-removing solvents can be used if bare lead ammo was fired thru it.
To clean other areas, such as the locking block, mag opening/chamber end, the slide engagement surfaces, slide, frame around the trigger bar, even the inside of the magazine tube, a nylon brush dipped in gun cleaning solvent is used.
The phosphor-bronze brush is for the bore, the nylon brush is for everywhere else.
Hope that helps!
*PS* If you use ultrasonics, there is no need to scrub with any kind of brush!