NSI or SSAIB — which is better?

Neither is better. Both are recognised UK inspectorate bodies, both are accepted by police forces for URN purposes, and a company certificated by either can arrange police response. What matters is that the certification is current, held by the entity you are contracting with, and covers the specific service you are buying.

The question comes up because installers use whichever body they belong to as a selling point, which implies a hierarchy that does not exist for this purpose.

The one nuance worth knowing is that NSI's Gold tier includes an audited quality management system, so it tells you something additional about how the business is run. That is a real difference, but it is a difference between NSI tiers as much as between the two bodies, and it is not what determines whether you can get police response.

Judge providers on the things that actually vary: whether the certification covers your service, who the monitoring ARC is, what the contract term is, what happens at renewal, and what the realistic response arrangement is for your address.