That's a really good link for generic answers to this question, as it comes up fairly frequently. Another way to think of the variable systems is just about everyone can do a Cat I so long as the weather required on the plate exists (except in Canada - later); many can do Cat II, but only with special training and special airport installations; few can do a Cat III primarily because there are very few Cat III installations (more in Europe than North America) but also because training, aircraft and ground equipment (as Jim referred to above) must also meet very high standards.
With regards visibility limits for an approach, in most of the world if the approach plate says 1/2 mile or 2600 RVR, then that is the minimum visibility that must be present to carry out and complete a Cat I ILS. In Canada however, these visibility listings are advisory only. The pilot may do the approach and if the "required visibility reference" is there at DH, then he/she can land. If not, a Go Around must be executed. Only if the visibility goes below 1/4 mile AND the RVR goes below 1200' must the pilot refuse the approach clearance. Oddly enough, the same approach ban (1/4 mile AND 1200 RVR) apply to completing a Cat II approach.
Many ask: Wherein lies the difference? It is a question that is being addressed by the regulators much to the consternation of many Canadian operators. There's a good chance a Cat I ILS will be restricted to 1800 RVR very shortly, and non-precision approaches to 1 mile.
But that's food for another thread!