G-10 is a generic designation for a glass-cloth composite, and Micarta is technically a trademarked name for a line of composites made by Norplex. However, "Micarta" has gotten to the point of being a generic trademark (like Kleenex, Velcro, Escalator, etc.), with the public calling any cloth-based epoxy composite "micarta" regardless of who made it. There is also the possibility for overlap, since Norplex-Micarta actually makes G-10 composites...
Because the terms are used generically in the knife industry, both have a wide array of quality. Some G-10 feels especially "plasticy" while some is much nicer. Micarta, because of the different substrate materials used, can be all over the place. Micarta can sometimes feel fuzzy, which some folks don't like; manufacturers might specifically choose to give their micarta that sort of finish, which can increase grip. G-10, because the substrate is nonporous, can be more resistant to liquids and staining than micarta.
Since the difference is mostly aesthetic, my preference will change based on the application. In my experience, polished micarta feels nicer than polished G-10, so I'd go with micarta for applications where that's the desired finish.