I don't understand what you mean by regular bleach flour?
Generally speaking, there are three types of wheat flour used in America: Pastry flour, all-purpose flour, and bread flour. They are defined by the percentage of gluten-forming proteins. Sometimes their intended purpose determines the fineness of the grind as well.
Pastry flour (which is also ground very fine) is made from soft wheat, and has a comparatively low protein level. Bread flour is high in protein (as is it's related "high gluten flour"), ground from hard winter wheat. All purpose is, for most practical purposes, a combination of the two.
There are, to be sure, geographic differences. In the South, for instance, all flours tend to be softer than in the rest of the country. And there are some differences depending on the particular mill. KA unbleached all-purpose flour, for instance, is said to be equal to Gold Medal bread flour. And that could be why your cookies came out different; a difference, as KCZ points out, in protein levels.
Bleached or unbleached has to do with an additional process, which, among other things, removes the beta-carotene content of the flour (and, with it, some of the nutritional value). Unbleached is particularly important for bread making, because the beta-carotene adds both color and a subtle flavor difference.
For cookies, bleached or unbleached, per se, shouldn't make a significant difference in the outcome.
The important thing is to realize that not all flours are created equal. Once you find one, for a particular purpose, that you are happy with, it's best to stick with it.