Does their 'separation' mean something about spacing on finals and runway vacating time?
Yes, in this case Schiphol ATC is referring to increased
in-flight separation on final, as required to protect the ILS signal. When you're shooting a Cat II/III approach, you don't want a big A380 in front of you on the same localizer potentially blocking the signal! This signal masking is one of the weaknesses of ILS when compared to GPS based approaches.
In ICAO land as weather deteriorates, different LVP phases come into effect at the airport (e.g., Phases A, B, C and D).
For example, LVP Phase A might equate to CAT I conditions with few operational restrictions. The airport might stop some maintenance activities, restrict ground vehicle movements, etc.
If conditions worsen, the airport might invoke LVP Phase B procedures, which might equate to CAT II standards, increased in-flight separation, slowing down the rate of arrivals & departures, restrictions on which runways to use, etc.
With very poor RVR the airport might declare LVP Phase D, with only one CAT III runway for landings, max in-flight separation on final, huge caps on arrivals & departure rates, mandatory use of the follow-me truck or towing for taxi, etc.
(These phases and associated restrictions will differ for each airport).
So back to the original topic, ATC can't offer CAT III service if they are applying in-flight separation and associated flow-control for CAT II.