Do separate neural circuits express separate genes in the same type of brain matter? In other words, even though there may be two distinct neural circuits which process separate forms of information in the same type of brain matter (i.e. cerebral cortex processing language in one area and planning in another area) can gene expression differences be found between such distinct and separate neural circuits, despite being in the same type of brain matter?
So, I am revisiting your question as asked by community, and I discover I misunderstood it initially: you didn't say neurons process information, you said neural circuits do. And that distinct neural circuits process separate forms of information. Well, that is inexact. There are several types of neurons, adapted to create neural circuits able to process motor, sensory and a combination of the two. Here's a jolly picture of some of them from here   : 


As for your question itself, it is quite confusing, at least at the level of knowledge of the brain we have at the moment: Do separate neural circuits express separate genes? We hardly know the surface of how the brain works, we consider a great improvement in science the fact that the psychotic spectrum gene pool has been somewhat mapped . 

However, to somewhat answer your question, for example,  there is a gene  that is sometimes (not always) involved in Tourette syndrome and has been studied. The main idea is that "The SLITRK1 protein probably plays a role in the development of nerve cells, including the growth of specialized extensions (axons and dendrites) that allow each nerve cell to communicate with nearby cells. " Now, Tourette's  might be associated with Altered parvalbumin-positive neuron distribution in basal ganglia, which could be either the result of SLITRK1 gene mutation, or some other gene. 

Let us put it this way: we do not have genes that suddenly create a third arm or a second heart, so we shouldn't have genes that suddenly create the wrong kind of neurons in the wrong region of the brain (I suppose that's what you mean by "type of brain matter", region of the brain, as neurons are generally all on the same level of the brain matter). 

My best hunch, judging by the research I presented above, is that gene mutations might encourage or discourage the development of the type of neurons you'd expect in certain areas, which improves or disables the ability to process this or that type of information. Same as above, expressions of the genes involved in making this or that protein or physiological process that favors development of neurons in very specific areas of the brain - will also influence the way or intensity we process information with that specific region of the brain. 

But - and it's a revelation I had myself while documenting this answer - genes do not express or suppress neuron development directly. The brain is what it is, and has the same format , if you will, no matter whose you dissect. 

Genes influence metabolic and general physiologic process that are also part of the brain development. Your brain regions will process better or poorer due to the types of proteins that trigger development for the neurons, neurotransmitters that control the electric flow through the etc involved in the matter. Neural cells are hard-coded. Their development and functioning aren't.