In terms of aircraft equipment, approach minimums, procedural differences, and anything else relevant, how do the 3 types of a Cat III ILS differ from one another?
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.
FAA Reference Material
The below links are to comprehensive FAA publications covering the areas as titled.
Thanks @Sports Racer for the comment with links to these documents.