Specular gives you highlights based on your lighting. It can also be a color. Specular halo is an alpha value or grayscale that controls the fade of the reflected lights.

Here's an image to explain it.
The first ball the colors of specular and specular halo are black. Result No reflections
The Second ball specular kicked up to white. You get a little dot where the sun is.
The third ball specular and halo set to white. You get a great big spot on half the surface.
The Forth ball A color used for specular, halo set to gray. You get a green dot with a little fade.
The fifth ball specular still green, halo driven by an image.This is great in Poser not so much in Bryce. If the halo areas don't line up with your lighting you can chop the reflection in half. That being said, with a light dome or IBL you can purposfully highlight specific parts of your object by painting specular and halo maps.