If you have glass-replacement coverage in your car's insurance policy, or you otherwise don't care how much a replacement windshield costs, then this article will be of academic interest only. But if you're in a position where you want to buy an inexpensive replacement windshield, then these issues are of importance.
First, the definitions: OEM means "original equipment manufacturer". This term is used in all industries to mean the company that makes the component, whatever it is, that goes into the original product, as sold by the company that "owns" the original product. Anything that replaces any component of the original product is an aftermarket part. This applies even when the replacement part is supplied by the same OEM that made the original one! It makes sense that the OEM's would sell to the aftermarket rather than just to the original market, but they don't have a monopoly; there are "pure" aftermarket manufacturers (AMM's) which supply the same parts.
The primary reason for getting an aftermarket windshield instead of an OEM is price. All windshields in the U.S. must conform to five Federal Motor Vehicle Safety Standards:
Given all this, what else would the difference be? OEMs work with the car's manufacturer very closely and set the dimensions and clearances of the windshield very precisely, while AMM's have to reverse- engineer their products. A common complaint about aftermarket windshields is that they're noisy — wind whistles or rattling; this would be why. But that, again, is due to low labor costs and cheap equipment in many offshore locations, and that is also changing as the market spreads world-wide. Laser-measuring tools can define windshields so precisely these days that they can't be told from the OEMs.
It would seem that the best way to proceed is to stay, if possible, with an OEM windshield until such time as the AMMs match them in price and quality. That is a matter of research — word of mouth, the Internet, the government, and the glass industry itself. A reputable windshield-repair shop will have no qualms about giving you the facts, and may be able to install an aftermarket windshield for you if their company affiliation permits it. But, as in everything else, "caveat emptor" — let the buyer beware!