What is different between a Cat IIIA, Cat IIIB, and Cat IIIC ILS approach?
ICAO and FAA CAT III definitions

A CAT III operation is a precision approach at lower than CAT II minima. Sub-categories are listed below.


A category III A approach is a precision instrument approach and landing
with no decision height or a decision height lower than 100ft (30m) and a runway
visual range not less than 700ft (200m).
A category III B approach is a precision approach and landing with no
decision height or a decision height lower than 50ft (15m) and a runway visual
range less than 700ft (200m), but not less than 150ft (50m).
A category III C approach is a precision approach and landing with no
decision height and no runway visual range limitation.


*I've omitted the JAA definitions.

Source Airbus Flight Operations Support documentation.



FAA Reference Material

The below links are to comprehensive FAA publications covering the areas as titled. 

FAA AC120-29 for CAT I/II. 

FAA AC120-28 for CAT III. 

Thanks @Sports Racer for the comment with links to these documents.