It seems to me the difference is not so subtle. As I understand from https://commonsclause.com/ you can use Commons Clause licensed code-library as part of your commercial application without having to make your source-code available whereas with AGPL you would have to. But I may be wrong?
Commons Clause is added ontop of an exisiting FOSS license. So it will be whatever requirements the base license plus an anti-commercial restriction preventing others from offering SaaS services.
